Torus Actions and Their Applications in Topology and Combinatorics

Home , Torus

Torus actions and their applications in top ology

and combinatorics

Victor M Buchstab er

Taras E Panov

Author address

Department of Mathematics and Mechanics Moscow State Univer

sity Moscow RUSSIA

Email address buchstabmechmathmsusu

Department of Mathematics and Mechanics Moscow State Univer

sity Moscow RUSSIA

Email address tpanovmechmat hmsusu

Mathematics Subject Classication B Q R M B

F C

Abstract Here the study of torus actions on top ological spaces is presented

connecting combinatorial and convex geometry with commutative and as a bridge

homological algebra algebraic geometry and top ology This link helps in under

standing the geometry and top ology of a space with torus action by studying the

combinatorics of the space of orbits Conversely the most subtle prop erties of a

combinatorial ob ject can b e recovered by realizing it as the orbit structure for a

prop er manifold or complex acted on by a torus The latter can b e a symplectic

manifold with Hamiltonian torus action a toric variety or manifold a subspace

arrangement complement etc while the combinatorial ob jects include simplicial

and cubical complexes p olytop es and arrangements This approach also provides

a natural top ological interpretation in terms of torus actions of many constructions

from commutative and homological algebra used in combinatorics

The exp osition centers around the theory of momentangle complexes provid

ing an eectiv e way to study triangulations by metho ds of equivariant top ology

The b o ok includes many new and wellknown op en problems and would b e suitable

e hop e that it will b e useful for sp ecialists b oth in top ology and as a textb o ok W

in combinatorics and will help to establish even tighter connections b etween the

sub jects involved

Contents

Intro duction

olytop es Chapter P

Denitions and main constructions

Face vectors and DehnSommerville equations

theorem The g

Upp er Bound and Lower Bound theorems

StanleyReisner face rings of simple p olytop es

Chapter Top ology and combinatorics of simplicial complexes

Abstract simplicial complexes and p olyhedrons

Basic P L top ology and op erations with simplicial complexes

Simplicial spheres

Triangulated manifolds

Bistellar moves

Chapter Commutative and homological algebra of simplicial complexes

StanleyReisner face rings of simplicial complexes

CohenMacaulay rings and complexes

Homological algebra background

Homological prop erties of face rings Toralgebras and Betti numb ers

Gorenstein complexes and DehnSommerville equations

Chapter Cubical complexes

Denitions and cubical maps

Cubical sub divisions of simple p olytop es and simplicial complexes

Chapter Toric and quasitoric manifolds

Toric varieties

Quasitoric manifolds

Stably complex structures and quasitoric representatives in cob ordism

classes

Combinatorial formulae for Hirzebruch genera of quasitoric manifolds

Classication problems

Chapter Momentangle complexes

Momentangle manifolds Z dened by simple p olytop es

General momentangle complexes Z

Cell decomp ositions of momentangle complexes vii

viii CONTENTS

Momentangle complexes corresp onding to joins connected sums and

bistellar moves

Borel constructions and DavisJan uszkiewicz space

Walk around the construction of Z generalizations analogues and

additional comments

Chapter Cohomology of momentangle complexes and combinatorics of

triangulated manifolds

b ergMo ore sp ectral sequence The Eilen

Cohomology algebra of Z

Bigraded Betti numb ers of Z the case of general K

Bigraded Betti numb ers of Z the case of spherical K

Partial quotients of Z

Bigraded Poincare duality and DehnSommerville equations

Chapter Cohomology rings of subspace arrangement complements

General arrangements and their complements

Co ordinate subspace arrangements and the cohomology of Z

Diagonal subspace arrangements and the cohomology of Z

Bibliography

Index

Intro duction

Torus actions on top ological spaces is classical and one of the most develop ed elds

in equivariant top ology Sp ecic problems connected with torus actions arise in dif

ferent areas of mathematics and mathematical physics which results in p ermanent

interest in the theory constant source of new applications and p enetration of new

ideas in top ology Many volumes devoted to particular asp ects of this wide eld

of mathematical knowledge are available The top ological approach is the sub ject

of monograph by G Bredon Monograph by M Audin deals with torus

actions from the symplectic geometry viewp oint The algebrogeometrical part of

geometry is the study known as the geometry of toric varieties or simply toric

presented in several texts These include V Danilovs original survey article

and more recent monographs by T Oda W Fulton and G Ewald

The orbit space of a torus action carries a rich combinatorial structure In many

cases studying the combinatorics of the quotient is the easiest and the most ecient

way to understand the top ology of a toric space This approach works in the op

p osite direction as well the equivariant top ology of a torus action sometimes helps

to interpret and prove the most subtle combinatorial results top ologically In the

most symmetric and regular cases such as pro jective toric varieties or Hamiltonian

torus actions on symplectic manifolds the quotient can b e identied with a convex

p olytop e More general toric spaces give rise to other combinatorial structures re

lated with their quotients Examples here include simplicial spheres triangulated

manifolds general simplicial complexes cubical complexes subspace arrangements

etc

Combined applications of combinatorial top ological and algebrogeometrical

metho ds stimulated intense development of toric geometry during the last three

decades This remarkable conuence of ideas enriched all the sub jects involved

with a numb er of sp ectacular results Another source of applications of top ological

and algebraic metho ds in combinatorics was provided by the theory of Stanley

Reisner face rings and CohenMacaulay complexes describ ed in R Stanleys mono

Our motivation was to broaden the existing bridge b etween torus graph

actions and combinatorics by giving some new constructions of toric spaces which

naturally arise from combinatorial considerations We also interpret many exist

ing results in such a way that their relationships with combinatorics b ecome more

transparent Traditionally simplicial complexes or triangulations were used in

top ology as a to ol for combinatorial treatment of top ological invariants of spaces

or manifolds On the other hand triangulations themselves can b e regarded as

particular structures so the space of triangulations b ecomes the ob ject of study

The idea of considering the space of triangulations of a given manifold has b een

INTRODUCTION

also motivated by some physical problems One gets an eective way of treat

ment of combinatorial results and problems concerning the numb er of faces in a

triangulation by interpreting them as extremal value problems on the space of tri

angulations We implement some of these ideas in our b o ok as well by constructing

and investigating invariants of triangulations using the equivariant top ology of toric

spaces

The b o ok is intended to b e a systematic but elementary overview for the asp ects

of torus actions mostly related to combinatorics However our level of exp osition

is not balanced b etween top ology and combinatorics We do not assume any par

ticular readers knowledge in combinatorics but in top ology a basic knowledge of

characteristic classes and sp ectral sequences techniques may b e very helpful in the

last chapters All necessary information is contained for instance in S No vikovs

b o ok We would recommend this b o ok since it is reasonably concise has

a rather broad scop e and pays much attention to the combinatorial asp ects of

top ology Nevertheless we tried to provide necessary background material in the

algebraic top ology and hop e that our b o ok will b e of interest to combinatorialists

as well

A signicant part of the text is devoted to the theory of momentangle com

plexes currently b eing develop ed by the authors This study was inspired by

pap er of M Davis and T Januszkiewicz where a top ological analogue of toric

varieties was intro duced In their work Davis and Januszkiewicz used a certain

space Z assigned to every simplicial complex on the vertex set universal T

m f mg In its turn the denition of Z was motivated see x by

the construction of the Coxeter complex of a Coxeter group and its generalizations

by E Vinb erg

Our approach brings the space Z to the center of attention To each subset

m k mk

m there is assigned a canonical T equivariant emb edding D T

m m m

D where D is the standard p olydisc in C and k is the cardinality of

This corresp ondence extends to any simplicial complex K on m and pro duces a

canonical bi graded cell decomp osition of the DavisJanuszkiewicz T space Z

which we refer to as the momentangle complex There is also a more general version

of momentangle complexes dened for any cubical sub complex in a unit cub e see

section The construction of Z gives rise to a functor see Prop osition

from the category of simplicial complexes and inclusions to the category of T

spaces and equivariant maps This functor induces a homomorphism b etween the

standard simplicial chain complex of a simplicial pair K K and the bigraded

cellular chain complex of Z Z The remarkable prop erty of the functor is

K K

1 2

that it takes a simplicial Lefschetz pair K K ie a pair such that K n K is an

op en manifold to another Lefschetz pair of momentangle complexes in such a

way that the fundamental cycle is mapp ed to the fundamental cycle For instance

if K is a triangulated manifold then the simplicial pair K is mapp ed to the

T and Z n Z is an op en manifold Studying the pair Z Z where Z

K K

functor K Z one interprets the combinatorics of simplicial complexes in terms

of the bigraded cohomology rings of momentangle complexes In the case when K

is a triangulated manifold the imp ortant additional information is provided by the

bigraded Poincare duality for the Lefschetz pair Z Z For instance the duality

implies the generalized DehnSommerville equations for the numb ers of k simplices

in a triangulated manifold

INTRODUCTION

Each chapter and most sections of the b o ok refer to a separate sub ject and

contain necessary intro ductory remarks Below we schematically overview the con

tents The chapter dep endence chart is shown in Figure

1 2 4 6 7 8

Figure Chapter dep endence scheme

Chapter contains combinatorial and geometrical background material on con

vex p olytop es Since a lot of literature is available on this sub ject see eg recent

we just give a short overview of construc excellent lectures by G Ziegler

tions used in the b o ok Although most of these constructions descend from the

convex geometry we tried to emphasize their combinatorial prop erties Section

contains two classical denitions of convex p olytop es examples the notions of

simple and simplicial p olytop es and the construction of connected sum of simple

p olytop es In section we intro duce the f and the hvector of a p olytop e and

give a Morsetheoretical pro of of the DehnSommerville equations Section

is devoted to the g theorem and in section we discuss the Upp er Bound and

the Lower Bound for the numb er of faces of a simple or simplicial p olytop e In

section we intro duce the StanleyReisner ring of a simple p olytop e

Simplicial complexes app ear in the full generality in Chapter In section

we dene abstract and geometrical simplicial complexes p olyhedrons In sec

tion we intro duce some standard notions from PLtop ology and describ e basic

constructions of simplicial complexes joins connected sums etc We also dis

cuss the Alexander duality and its simplicial version here From the early days

of top ology triangulations of nice top ological spaces such as manifolds or spheres

were of particular interest Triangulations of spheres or simplicial spheres are

the sub ject of section Here we also discuss the interrelations b etween some

particular sub classes of simplicial spheres such as PLspheres p olytopal spheres

etc and one famous combinatorial problem the socalled g conjecture for face

vectors Triangulated or simplicial manifolds are the sub ject of section some

related op en problems from lowdimensional and PL top ology are also included

there The notion of bistellar moves as a particularly interesting and useful class

of op erations on simplicial complexes is discussed in section

In chapter we give an overview of commutative algebra involved in the combi

natorics of simplicial complexes Many of the constructions from this chapter esp e

cially those app earing in the b eginning are taken from Stanleys monograph

however we tried to emphasize their functorial prop erties and relationships with

op erations from chapter The StanleyReisner face ring of simplicial complex is

intro duced in section The imp ortant class of CohenMacaulay complexes is the

INTRODUCTION

sub ject of section we also give Stanleys argument for the Upp er Bound theo

rem for spheres here Section contains the homological algebra background in

cluding resolutions and the graded functor Tor Koszul complexes and Toralgebras

asso ciated with simplicial complexes are describ ed in section together with their

complexes are the sub ject of basic prop erties Gorenstein algebras and Gorenstein

section This class of selfdual CohenMacaulay complexes contains simplicial

and homology spheres and in a sense provides the b est p ossible algebraic approx

imation to them The chapter ends up with a discussion of some generalizations of

the DehnSommerville equations

Cubical complexes are the sub ject of chapter We give denitions and dis

cuss some interesting related problems from the discrete geometry in section

Section intro duces some particular cubical complexes necessary for the con

struction of momentangle complexes These include the cubical sub divisions of

simple p olytop es and simplicial complexes

Dierent asp ects of torus actions is the main theme of the second part of the

b o ok Chapter starts with a brief review of the algebraic geometry of toric va

rieties in section We stress up on those features of toric varieties which can

b e taken as a starting p oint for their subsequent top ological generalizations We

also give Stanleys famous argument for the necessity part of the g theorem one

of the rst and most known applications of the algebraic geometry in the combi

natorics of p olytop es In section we give the denition and basic prop erties

of quasitoric manifolds the notion intro duced by Davis and Januszkiewicz under

the name toric manifolds as a top ological generalization of toric varieties The

top ology of quasitoric manifolds is the sub ject of sections and this includes

the discussion of their cohomology cob ordisms characteristic classes Hirzebruch

genera etc Quasitoric manifolds work particularly well in the cob ordism theory

and may serve as a convenient framework for dierent cob ordism calculations Evi

dences for this are provided by some recent results of V Buchstab er and N Ray It

is proved that a certain class of quasitoric manifolds provides an alternative addi

tive basis for the complex cob ordism ring Note that the standard basis consists of

Milnor hyp ersurfaces which are not quasitoric Moreover using the combinatorial

construction of connected sum of p olytop es it is proved that each complex cob or

dism class contains a quasitoric manifold with a canonical stably almost complex

structure resp ected by the torus action Since quasitoric manifolds are necessarily

connected the nature of this result resembles the famous Hirzebruch problem ab out

connected algebraic representatives in complex cob ordisms All these arguments

presented in section op en the way to evaluation of global cob ordism invari

ants on manifolds by cho osing a quasitoric representative and studying the lo cal

invariants of the action As an application in section we give combinatorial

formulae due to the second author for Hirzebruch genera of quasitoric manifolds

Section is a discussion of several known results on the classication of toric and

quasitoric manifolds over a given simple p olytop e

The theory of momentangle complexes is the sub ject of chapters and We

start in section with the denition of the momentangle manifold Z corre

sp onding to a simple p olytop e P The general momentangle complexes Z are

intro duced in section using sp ecial cubical sub divisions from section Here

we prove that Z is a manifold provided that K is a simplicial sphere Two typ es

of bigraded cell decomp ositions of momentangle complexes are intro duced in sec

tion In section we discuss dierent functorial prop erties of momentangle

INTRODUCTION

complexes with resp ect to simplicial maps and constructions from section A

basic homotopy theory of momentangle complexes is the sub ject of section

Concluding section aims for a more broad view on the constructions of qua

sitoric manifolds and momentangle complexes We discuss dierent interrelations

similar constructions and p ossible generalizations there

The cohomology of momentangle complexes and its r ole in investigating com

binatorial invariants of triangulations is studied in chapter In section

we review the Eilenb ergMo ore sp ectral sequence our main computational to ol

The bigraded cellular structure and the Eilenb ergMo ore sp ectral sequence are the

main ingredients in the calculation of cohomology of a general momentangle com

plex Z carried out in section Additional results on the cohomology in the

case when K is a simplicial sphere are given in section These calculations reveal

some new connections with wellknown constructions from homological algebra and

op en the way to some further combinatorial applications In particular the coho

mology of the Koszul complex for a StanleyReisner ring and its Betti numb ers now

get a top ological interpretation In section we study the quotients of moment

T of rank m Quasitoric manifolds arise angle manifolds Z by subtori H

in this scheme as quotients for freely acting subtori of the maximal p ossible rank

Momentangle complexes corresp onding to triangulated manifolds are considered

in section In this situation all singular p oints of Z form a single orbit of the

torus action and the complement of an equivariant neighb orho o d of this orbit is a

manifold with b oundary

In chapter we apply the theory of momentangle complexes to the top ology

of subspace arrangement complements Section is a brief review of general ar

rangements Then we restrict to the cases of co ordinate subspace arrangements

and diagonal subspace arrangements sections and resp ectively In partic

ular we calculate the cohomology ring of the complement of a co ordinate subspace

arrangement by reducing it to the cohomology of a momentangle complex This

also reveals some remarkable connections b etween certain results from commutative

algebra of monomial ideals such as the famous Ho chsters theorem and top olog

ical results on subspace arrangements eg the GoreskyMacpherson formula for

the cohomology of complement In the diagonal subspace arrangement case the

cohomology of complement is included as a canonical subspace into the cohomology

of the lo op space on a certain moment angle complex Z

Almost all new concepts in our b o ok are accompanied with explanatory exam

ples We also give many examples of particular computations illustrating general

theorems Throughout the text the reader will encounter a numb er of op en prob

lems Some of these problems and conjectures are widely known while others are

new In most cases we tried to give a top ological interpretation for the question

under consideration which might provide an alternative approach to its solution

Many of those results in the b o ok which are due to the authors have already

app eared in their pap ers or pap ers by N Ray and

the rst author We sometimes omit the corresp onding quotations in the text The

whole b o ok has grown up from our survey article

wledgements Both authors are indebted to Sergey Novikov whose Ackno

inuence on our top ological education cannot b e overestimated The rst author

takes this opp ortunity to express sp ecial thanks to Nigel Ray for the very pleasant

joint work during the last ten years which in particular generated some of the

INTRODUCTION

ideas develop ed in this b o ok The second author also wishes to express his deep

gratitude to N Ray for extremely fruitful collab oration and sincere hospitality

during his stay in Manchester The authors wish to thank Levan Alania Y usuf

Civan Natalia Dobrinskaya Nikolai Dolbilin Mikhail Farb er Konstantin Feldman

Ivan Izmestiev Frank Lutz Oleg Musin Andrew Ranicki Elmer Rees Mikhail

Shtanko Mikhail Shtogrin Vladimir Smirnov James Stashe Neil Strickland

unter Sergey Tarasov Victor Vassiliev Volkmar Welker Sergey Yuzvinsky and G

Ziegler for the insight gained from discussions on the sub ject of our b o ok

We are also thankful to all participants of the seminar Top ology and Com

putational Geometry which is b eing held by O R Musin and the authors at the

Department of Mathematics and Mechanics Lomonosov Moscow State University

We would like to gratefully acknowledge the most helpful comments correc

tions and additional references suggested by the referees Thanks to them the text

has b een signicantly enhanced in many places

The work of b oth authors was partially supp orted by the Russian Founda

tion for Fundamental Research grant no and the Russian Leading

Scientic Scho ol Supp ort grant no The second author was also sup

p orted by the British Royal So cietyNA TO Postdo ctoral Fellowship while visiting

the Department of Mathematics at the University of Manchester

CHAPTER

olytop es P

Denitions and main constructions

Both combinatorial and geometrical asp ects of the theory of convex p olytop es

are exp osed in a vast numb er of textb o oks monographs and pap ers Among

them are the classical monograph by Grun baum and more recent Zieglers lec

tures Face vectors and other combinatorial questions are discussed in b o oks

YemelichevKovalevKravtsov by McMullenShephard Brnsted

and survey article by Klee and Kleinschmidt These sources contain a host of

further references In this section we review some basic concepts and constructions

used in the rest of the b o ok

There are two algorithmically dierent ways to dene a convex p olytop e in

dimensional ane Euclidean space R n

polytope is the convex hull of a nite set of p oints Definition A convex

in some R

Definition A convex polyhedron P is an intersection of nitely many

halfspaces in some R

i P x R hl x i > a m

i i

are some linear functions and a R i m A convex where l R

i i

polytope is a b ounded convex p olyhedron

Nevertheless the ab ove two denitions pro duce the same geometrical ob ject

ie the subset of R is a convex hull of a nite p oint set if and only if it is a

b ounded intersection of nitely many halfspaces This classical fact is proved in

many textb o oks on p olytop es and convex geometry see eg Theorem

Definition The dimension of a p olytop e is the dimension of its ane

hull Unless otherwise stated we assume that any ndimensional p olytop e or sim

n n

ply npolytope P is a subset in ndimensional ambient space R A supporting

n n

hyperplane of P is an ane hyp erplane H which intersects P and for which the

p olytop e is contained in one of the two closed halfspaces determined by the hyp er

H is then called a face of the p olytop e We also regard plane The intersection P

the p olytop e P itself as a face other faces are called proper faces The boundary

n n

P is the union of all prop er faces of P Each face of an np olytop e is itself a

p olytop e of dimension 6 n dimensional faces are called vertices dimensional

faces are edges and co dimension one faces are facets

n n

1 2

Two p olytop es P R and P R of the same dimension are said to b e

n n

1 2

equivalent or anely isomorphic if there is an ane map R R anely

that is a bijection b etween the p oints of the two p olytop es Two p olytop es are

combinatorial ly equivalent if there is a bijection b etween their sets of faces that

preserves the inclusion relation

POLYTOPES

Note that two anely isomorphic p olytop es are combinatorially equivalent but

the opp osite is not true A more consistent denition of combinatorial equivalence

uses the combinatorial notions of p oset and lattice

Definition A poset or nite partially ordered set S 6 is a nite set

S equipp ed with a relation 6 which is reexive x 6 x for all x S transitive

x 6 y and y 6 z imply x 6 z and antisymmetric x 6 y and y 6 x imply x y

When the partial order is clear we denote the p oset by just S A chain in S is a

totally ordered subset of S

Definition The faces of a p olytop e P of all dimensions form a p oset with

resp ect to inclusion called the face poset

Now we observe that two p olytop es are combinatorially equivalent if and only

if their face p osets are isomorphic More information ab out face p osets of p olytop es

can b e found in x

Definition A combinatorial polytope is a class of combinatorial equiva

lent convex or geometrical p olytop es Equivalently a combinatorial p olytop e is

the face p oset of a geometrical p olytop e

Agreement Supp ose that a p olytop e P is represented as an intersection

of halfspaces as in In the sequel we assume that there are no redundant

inequalities hl x i > a in such a representation That is no inequality can b e

i i

n n

removed from without changing the p olytop e P In this case P has exactly

m facets which are the intersections of hyp erplanes hl x i a i m

i i

with P The vector l is orthogonal to the corresp onding facet and p oints towards

the interior of the p olytop e

simplex and cub e An ndimensional simplex Example is the con

p oints in R that do not lie on a common ane hyp erplane vex hull of n

All faces of an nsimplex are simplices of dimension 6 n Any two nsimplices are

anely equivalent The standard nsimplex is the convex hull of p oints

and in R Alternatively the standard n

simplex is dened by n inequalities

x > i n and x x >

i n

The regular nsimplex is the convex hull of n p oints

in R

q q

The standard q cube is the convex p olytop e I R dened by

q q

I fy y R 6 y 6 i q g

q i

q q

Alternatively the standard q cub e is the convex hull of the p oints in R that

have only zero or unit co ordinates

The following construction shows that any convex np olytop e with m facets is

anely equivalent to the intersection of the positive cone

m m m

R y y R y > i m R

m i

with a certain ndimensional plane

uction Let P R b e a convex np olytop e given by with Constr

some l R a R i m Form the n mmatrix L whose columns

i i

are the vectors l written in the standard basis of R ie L l Note that

i j i i j

DEFINITIONS AND MAIN CONSTRUCTIONS

t m

L is of rank n Likewise let a a a R b e the column vector with

entries a Then we can rewrite as

n t

P x R L x a > i m

t t

where L is the transp osed matrix and x x x is the column vector

Consider the ane map

n m t m

A R R A x L x a R

P P

Its image is an ndimensional plane in R and A P is the intersection of this

plane with the p ositive cone R see Let W b e an m m nmatrix whose

columns form a basis of linear dep endencies b etween the vectors l That is W is

a rank m n matrix satisfying LW Then it is easy to see that

m t t

A P y R W y W a y > i m

P i

By denition the p olytop es P and A P are anely equivalent

n n

Example Consider the standard nsimplex R dened by inequali

ties It has m n facets and is given by with l

t t

l l a a a One can

n n n n

t t t

take W in Construction Hence W y y y W a

and we have

n n

A y R y y y > i n

n i

This is the regular nsimplex in R

The notion of generic p olytop e dep ends on the choice of denition of convex

p olytop e Below we describ e the two p ossibilities

A set of m n p oints in R is in general position if no n of them lie on

a common ane hyp erplane Now Denition implies that a convex p olytop e is

generic if it is the convex hull of a set of general p ositioned p oints This implies

that all prop er faces of the p olytop e are simplices ie every facet has the minimal

numb er of vertices namely n Such p olytop es are called simplicial

On the other hand a set of m n hyp erplanes hl x i a l R

i i i

x R a R i m is in general position if no p oint b elongs to more

than n hyp erplanes From the viewp oint of Denition a convex p olytop e P

is generic if its b ounding hyp erplanes see are in general p osition That is

there are exactly n facets meeting at each vertex of P Such p olytop es are called

simple Note that each face of a simple p olytop e is again a simple p olytop e

Definition For any convex p olytop e P R dene its polar set P

R by

fx hx x i > for all x P g P R

Remark We adopt the denition of the p olar set from the algebraic geometry

of toric varieties not the classical one from the convex geometry The latter is

obtained by replacing the inequality > ab ove by 6 Obviously the toric

geometers p olar set is taken into the convex geometers one by the central symmetry

with resp ect to

It is well known in convex geometry that the p olar set P is convex in the dual

space R and is contained in the interior of P Moreover if P itself contains

in its interior then P is a convex p olytop e ie is b ounded and P P

POLYTOPES

see eg x The p olytop e P is called the polar or dual of P There is

a onetoone order reversing corresp ondence b etween the face p osets of P and P

In other words the face p oset of P is the opp osite of the face p oset of P In

particular if P is simple then P is simplicial and vice versa

Example Any p olygon p olytop e is simple and simplicial at the same

time In dimensions > the simplex is the only p olytop e that is simultaneously

simple and simplicial The cub e is a simple p olytop e The p olar of simplex is again

the simplex The p olar of cub e is called the crosspolytope The dimensional

crossp olytop e is known as the o ctahedron

Construction Pro duct of simple p olytop es The product P P of

two simple p olytop es P and P is a simple p olytop e as well The dual op eration

n n

1 2

on simplicial p olytop es can b e describ ed as follows Let S R and S R b e

two simplicial p olytop es Supp ose that b oth S and S contain in their interiors

Now dene

n n

1 2

S S conv S S R

here conv means the convex hull It is easy to see that S S is a simplicial

p olytop e and for any two simple p olytop es P P containing in their interiors

the following holds

P P P P

Obviously b oth pro duct and op erations are also dened on combinatorial p oly

top es in this case the ab ove formula holds without any restrictions

Construction Connected sum of simple p olytop es Supp ose we are

n n

given two simple p olytop es P and Q b oth of dimension n with distinguished ver

n n

and w resp ectively The informal way to get the connected sum P Q tices v

v w

n n n n

of P at v and Q at w is as follows We cut o v from P and w from Q

then after a pro jective transformation we can glue the rest of P to the rest of

n n n

Q along the new simplex facets to obtain P Q Below we give the formal

v w

denition following x

First we intro duce an np olyhedron which will b e used as a template for

the construction it arises by considering the standard n simplex in

the subspace fx x g of R and taking its cartesian pro duct with the rst

co ordinate axis The facets G of therefore have the form R D where D

r r r

n n

r n are the facets of Both and the G are divided into p ositive

and negative halves determined by the sign of the co ordinate x

n n

E and the facets of Q We order the facets of P meeting in v as E

meeting in w as F F Denote the complementary sets of facets by C and

n v

C those in C avoid v and those in C avoid w

w v w

We now cho ose pro jective transformations and of R whose purp ose is

P Q

n n

to map v and w to x resp ectively We insist that emb eds P in so as

to satisfy two conditions rstly that the hyp erplane dening E is identied with

the hyp erplane dening G for each r n and secondly that the images of

the hyp erplanes dening C meet in its negative half Similarly identies the

v Q

hyp erplane dening F with that dening G for each r n but the images of

r r

the hyp erplanes dening C meet in its p ositive half We dene the connected

n n n n

sum P Q of P at v and Q at w to b e the simple convex np olytop e

v w

determined by the images of the hyp erplanes dening C and C and hyp erplanes

v w

dening G r n It is dened only up to combinatorial equivalence r

FACE VECTORS AND DEHNSOMMERVILLE EQUATIONS

moreover dierent choices for either of v and w or either of the orderings for E

and F are likely to aect the combinatorial typ e When the choices are clear or

n n

their eect on the result irrelevant we use the abbreviation P Q

The related construction of connected sum P S of a simple p olytop e P and

a simplicial p olytop e S is describ ed in Example

Example If P is an m gon and Q is an m gon then P Q is an

m m gon

If b oth P and Q are nsimplices then P Q I the pro duct of

n simplex and segment In particular for n we get a triangular prism

More generally if P is an nsimplex then P Q is the result of cutting

v w

the vertex w from Q by a hyp erplane that isolate w from other vertices For more

relations b etween connected sums and hyp erplane cuts see x

Definition A simplicial p olytop e S is called k neighborly if any k ver

tices span a face of S Likewise a simple p olytop e P is called k neighb orly if any

k facets of P have nonempty intersection ie share a common co dimensionk

face Obviously every simplicial or simple p olytop e is neighb orly It can b e

shown Corollary see also Example b elow that if S is a k neighb orly

then S is an nsimplex This implies that any simplicial np olytop e and k

neighb orly simplicial p olytop e is a simplex However there exist simplicial

np olytop es with an arbitrary numb er of vertices which are neighb orly Such

p olytop es are called neighborly In particular there is a simplicial p olytop e dif

ferent from the simplex any two vertices of which are connected by an edge

Example neighb orly p olytop e Let P b e the pro duct of

two triangles Then P is a simple p olytop e and it is easy to see that any two

facets of P share a common face Hence P is neighb orly The p olar P is a

neighb orly simplicial p olytop e

More generally if a simple p olytop e P is k neighb orly and a simple p olytop e

P is k neighb orly then the pro duct P P is a mink k neighb orly simple

n n n n

p olytop e It follows that and provide examples of

neighb orly simplicial n and n p olytop es The following example gives a

neighb orly p olytop e with an arbitrary numb er of vertices

Example cyclic p olytop es Dene the moment curve in R by

n n n

x t t t t R x R R t

For any m n dene the cyclic polytope C t t as the convex hull of m

distinct p oints x t t t t on the moment curve It follows from

i m

the Vandermonde determinant identity that no n p oints on the moment curve

t t is a simplicial n b elong to a common ane hyp erplane Hence C

p olytop e It can b e shown see Theorem that C t t has exactly

m vertices x t the combinatorial typ e of cyclic p olytop e do es not dep end on the

sp ecic choice of the parameters t t and C t t is a neighb orly

m m

simplicial np olytop e We will denote the combinatorial cyclic np olytop e with m

vertices by C m

Face vectors and DehnSommerville equations

The notion of the f vector or face vector is a central concept in the combi

natorial theory of p olytop es It has b een studied there since the time of Euler

POLYTOPES

Definition Let S b e a simplicial np olytop e Denote by f the numb er

of idimensional faces of S The integer vector f S f f is called

the f vector of S We also put f The hvector of S is the integer vector

h h h dened from the equation

n n n

h t h t h t f t f

n n n

where g g h h i is called Finally the sequence g g g

i i i

the g vector of S

The f vector of a simple np olytop e P is dened as the f vector of its p olar

f P f P and similarly for the h and the g vector of P More explicitly

f P f f where f is the numb er of faces of P of co dimension i

n i

ie of dimension n i In particular f is the numb er of facets of P which

we usually denote mP or just m The agreement f is now justied by the

fact that P itself is a face of co dimension

Remark The denition of hvector may seem to b e unnatural at rst glance

However as we will see later the hvector has a numb er of combinatorialgeometri

cal and algebraic interpretations and in some situations is more convenient to work

with than the f vector

Obviously the f vector is a combinatorial invariant of P that is it dep ends

only on the face p oset For convenience we assume all p olytop es in this section to

b e combinatorial unless otherwise stated

Example Two dierent combinatorial simple p olytop es may have same

f vectors For instance let P b e the cub e and P b e the simple p olytop e with

triangular quadrangular and p entagonal facets see Figure Note that P

f P is dual to the cyclic p olytop e C from Denition Then f P

H P

Z Z

H Z Z

Z Z

Figure Two combinatorially nonequivalent simple p oly

top es with the same f vectors

The f vector and the hvector carry the same information ab out the p olytop e

and determine each other by means of linear relations namely

k n

X X

ni q

k i

h f f h k n

k i nk nq

nk k

q k

FACE VECTORS AND DEHNSOMMERVILLE EQUATIONS

n n

In particular h and h f f f By Eulers

n n

theorem

n n

f f f

which is equivalent to h h In the case of simple p olytop es Eulers

theorem admits the following generalization

Theorem DehnSommerville relations The hvector of any simple or

ie simplicial npolytope is symmetric

h h i n

i ni

The DehnSommerville equations can b e proved in many dierent ways We

give a pro of which uses a Morsetheoretical argument which rst app eared in

We will return to this argument in chapter

n n

oof of Theorem Let P R b e a simple p olytop e Cho ose a Pr

R R which is generic in the sense that it distinguishes the linear function

n n

vertices of P For this there is a vector in R such that x h x i The

assumption on implies that is parallel to no edge of P Now we can view

n n

as a height function on P Using we make the skeleton of P a directed

graph by orienting each edge in such a way that increases along it this can b e

done since is generic see Figure For each vertex v of P dene its index

ind

X B

ind

H B

ind

H B

Figure Oriented skeleton of P and index of vertex

indv as the numb er of incident edges that p oint towards v Denote the numb er

of vertices of index i by I i We claim that I i h Indeed each face of P

has a unique top vertex the maximum of the height function restricted to the

k n

face and a unique b ottom vertex the minimum of Let F b e a k face of P

n k

its top vertex Since P is simple there are exactly k edges of F meeting and v F

POLYTOPES

at v whence indv > k On the other hand each vertex of index q > k is the

F F

top vertex for exactly faces of dimension k It follows that f the numb er

of k faces can b e calculated as

f I q

q >k

Now the second identity from shows that I q h as claimed In

particular the numb er I q do es not dep end on At the same time since

ind v n ind v for any vertex v one has

h I q I n q h

nq q

Using we can rewrite the DehnSommerville equations in terms of the

f vector as follows

f f k n

k j

j k

The DehnSommerville equations were established by Dehn for n 6 in

and by Sommerville in the general case in in the form similar to

n n

2 1

b e simple p olytop es Each face of P P and P Example Let P

is the pro duct of a face of P and a face of P whence

f P P f P f P k n n

k i k i

Set

hP t h h t h t

Then it follows from the ab ove formula and that

hP P t hP thP t

Example Let us express the f vector and the hvector of the connected

n n n n

sum P Q in terms of that of P and Q From Construction we deduce

that

n n n n

f P Q f P f Q i n

i i i

n n n n

f P Q f P f Q

n n n

Then it follows from that

n n n n

h P Q h P Q

n n n n

h P Q h P h Q i n

i i i

This prop erty raises the following question

Problem Describ e al l integervalued functions on the set of simple poly

topes which are linear with respect to the connected sum operation

The previous identities show that examples of such functions are provided by

h for i n i

THE g THEOREM

Example Supp ose S is a q neighb orly simplicial np olytop e see Deni

tion dierent from the nsimplex Then f S k 6 q From

we get

ni m mnk

k i

h S k 6 q

k i i k

The latter equality is obtained by calculating the co ecient of t from two sides of

the identity

m mnk

t t

Since S is not a simplex we have m n which together with gives

h h h These inequalities together with the DehnSommerville

mentioned in Denition equations imply the upp er b ound q 6

The g theorem

The g theorem gives answer to the following natural question which integer

vectors may app ear as the f vectors of simple or equivalently simplicial p oly

top es The DehnSommerville relations provide a necessary condition As far as

only linear equations are concerned there are no further restrictions

Proposition Klee The DehnSommervil le relations are the most

general linear equations satised by the f vectors of al l simple or simplicial poly

topes

Proof In the statement was proved directly in terms of f vectors How

ever the usage of hvectors signicantly simplies the pro of It is sucient to prove

that the ane hull of the hvectors h h h of simple np olytop es is an

simple p oly dimensional plane This can b e done for instance by providing

k nk

top es with anely indep endent hvectors Set Q k

k k

Since h t t the formula gives

k nk

t t

k nk

It follows that hQ hQ t t k

k k

Therefore the vectors h Q k are anely indep endent

Example Each vertex of a simple np olytop e P is contained in exactly

n edges and each edge connects two vertices This implies the following obvious

linear equation for the comp onents of the f vector of P

f nf

n n

Prop osition shows that this equation is a corollary of the DehnSommerville

equations One may observe that it is equation for k n It follows

from and Euler identity that for simple or simplicial p olytop es the

f vector is completely determined by the numb er of facets namely

f P f f f

POLYTOPES

We mention also that Euler identity is the only linear relation satised

by the face vectors of general convex p olytop es This can b e proved in a similar

way as Prop osition by sp ecifying suciently many p olytop es with anely

indep endent face vectors

The conditions characterizing the f vectors of simple or simplicial p olytop es

now know as the g theorem were conjectured by P McMullen in and

proved by R Stanley necessity and Billera and Lee suciency in

Besides the DehnSommerville equations the g theorem contains two groups

of inequalities one linear and one nonlinear To formulate the g theorem we need

the following construction

Definition For any two p ositive integers a i there exists a unique bi

nomial iexpansion of a of the form

a a a

i i1 j

i i j

where a a a > j > Dene

i i j

a a a

ii hii h

i i1 j

i i j

Example For a a

If i > a then the binomial expansion has the form

i i ia

hii

and therefore a a

Let a i The corresp onding binomial expansion is

Hence

Theorem g theorem An integer vector f f f is the f vec

tor of a simple npolytope if and only if the corresponding sequence h h

determined by satises the fol lowing three conditions

a h h i n the DehnSommervil le equations

i ni

b h 6 h 6 6 h i

hii

c h h h 6 h h i

i i i i

Remark Obviously the same conditions characterize the f vectors of simpli

cial p olytop es

Example The rst inequality h 6 h from part b of g theorem is

equivalent to f m > n This just expresses the fact that it takes at least

n hyp erplanes to b ound a p olytop e in R

Taking into account that

h n f f

see we rewrite the rst inequality h h 6 h h from part c of

g theorem as

n f n

nf f 6

see Example This is equivalent to the upp er b ound

f 6

THE g THEOREM

which says that two facets share at most one face of co dimension two In the dual

simplicial notations two vertices are joined by at most one edge

The second inequality h 6 h for n > from part b of g theorem is

equivalent to

f > nf

This is the rst and most signicant inequality from the famous L ower Bound

Conjecture for simple p olytop es see Theorem b elow

h (h +1)

1 1

h h h

2 2 1

Figure h h domain n >

Thus the rst two co ordinates of the hvectors of simple p olytop es P n >

h h

1 1

and h h in the h h plane see fall b etween the two curves h

Figure Note that the most general linear inequalities satised by p oints from

this domain are h > and h > h

Definition An integral sequence k k k satisfying k and

hii

for i r is called an M 6 k 6 k vector after M Macaulay

Conditions b and c from g theorem are equivalent to that the g vector

g g g of a simple np olytop e is an M vector The notion of M vector

arises in the following classication result of commutative algebra

Theorem An integral sequence k k k is an M vector if and

A A over a only if there exists a commutative graded algebra A A

eld k A such that

a A is generated as an algebra by degreetwo elements

b the dimension of ith graded component of A equals k

dim A k i r

k i

This theorem is essentially due to Macaulay but the ab ove explicit formulation

is that of The pro of can b e also found there

The pro of of the suciency part of g theorem due to Billera and Lee is quite

elementary and relies up on a remarkable combinatorialgeometrical construction of

a simplicial p olytop e with any prescrib ed M sequence as its g vector On the other

POLYTOPES

hand Stanleys pro of of the necessity part of g theorem ie that the g vector

of a simple p olytop e is an M vector used deep results from algebraic geometry

in particular the Hard Lefschetz theorem for the cohomology of toric varieties

We outline Stanleys pro of in section After several more elementary

combinatorial pro ofs of the g theorem app eared The rst such pro of is due to

McMullen It builds up on the notion of polytope algebra which substitutes the

cohomology algebra of toric variety Despite b eing elementary it was a complicated

pro of Later McMullen simplied his approach in Yet another elementary

pro of of the g theorem has b een recently found by Timorin It relies on

the interpretation of McMullens p olytop e algebra as the algebra of dierential

op erators with constant co ecients vanishing on the volume polynomial of the

p olytop e

Upp er Bound and Lower Bound theorems

The following statement now know as the Upper Bound Conjecture UBC

was suggested by Motzkin in and proved by P McMullen in

Theorem UBC for simplicial p olytop es F rom al l simplicial npolyto

pes S with m vertices the cyclic polytope C m Example has the maximal

number of ifaces 6 i 6 n That is if f S m then

f S 6 f C m for i n

i i

The equality in the above formula holds if and only if S is a neighborly polytope see

Denition

Note that since C m is neighb orly

f C m for i

Due to the DehnSommerville equations this determines the full f vector of C m

The exact values are given by the following lemma

The number of ifaces of cyclic polytope C m or any neigh Lemma

borly npolytope with m vertices is given by

n n

X X

q mnq np mnp

f C m i n

ni q ip p

q p

q where we assume for p

n n

Proof Using the second identity from identity n the

DehnSommerville equations and we calculate

n n

X X X

q q q

f h h h

i nq q nq

ni ni ni

q q

n n

X X

q mnq np mnp

ni q ip p

q p

The ab ove pro of justies the following statement

UPPER BOUND AND LOWER BOUND THEOREMS

Corollary The UBC for simplicial polytopes Theorem is implied

by the fol lowing inequalities for the hvector of a simplicial polytope S with m

vertices

mni

i h S 6

This was one of the key observations in McMullens original pro of of the UBC

for simplicial p olytop es see also x and x The ab ove corollary

is also useful for a dierent generalization of UBC we will return to this in sec

tion We note also that due to the argument of Klee and McMullen see

Lemma the UBT holds for al l convex p olytop es not necessarily simplicial

That is the cyclic p olytop e C m has the maximal numb er of ifaces from all

convex np olytop es with m vertices

Another fundamental fact from the theory of convex p olytop es is the Lower

Bound Conjecture LBC for simplicial p olytop es

Definition A simplicial np olytop e S is called stacked if there is a se

quence S S S S of np olytop es such that S is an nsimplex and S is

k i

obtained from S by adding a pyramid over some facet of S In the combinatorial

i i

language stacked p olytop es are those obtained from a simplex by applying several

subsequent stellar sub divisions of facets

Remark Adding a pyramid or stellar sub division of a facet is dual to cut

ting a vertex of a simple p olytop e see Example

Theorem LBC for simplicial p olytop es For any simplicial npolytope

S n > with m f vertices hold

n n

f S > f i for i n

i i

f S > n f n n

The equality is achieved if and only if S is a stacked polytope

The argument by McMullen Perles and Walkup reduces the LBC to the

case i namely the inequality f > f The LBC was rst proved by

Barnette The only if part in the statement ab out the equality was

proved in using g theorem Unlike the UBC little is know ab out generaliza

tions of the LBC theorem to nonsimplicial convex p olytop es Some results in this

direction were obtained in along with generalizations of the LBC theorem to

simplicial spheres and manifolds see also sections in this b o ok

In dual notations the UBC and the LBC provide upp er and lower b ounds for

the numb er of faces of simple p olytop es with given numb er of facets Both theorems

were proved approximately at the same time in and motivated P McMullen

to conjecture the g theorem On the other hand b oth UBC and LBC are

corollaries of the g theorem see eg x In fact the LBC follows only from

parts a and b of Theorem while the UBC follows from parts a and c

Part b of g theorem namely the inequalities

h 6 h 6 6 h

where suggested in as a generalization of the LBC for simplicial p olytop es

The second inequality h 6 h is equivalent to the i case of LBC see Exam

ple It follows from the results of and that are the strongest

POLYTOPES

p ossible linear inequalities satised by the f vectors of simple or simplicial p oly

top es compare with the comment after Example These inequalities are now

known as the Gener alized Lower Bound Conjecture GLBC

During the last two decades a lot of work was done in extending the Dehn

Sommerville equations the GLBC and the g theorem to ob jects more general than

simplicial p olytop es However there are still many intriguing op en problems here

For more information see the rst section of survey article by Stanley and

section in this b o ok

face rings of simple p olytop es StanleyReisner

The only aim of this short section is to dene the StanleyReisner ring of

a simple p olytop e This fundamental combinatorial invariant will b e one of the

main characters in the next chapter However it is convenient for us to give it an

indep endent treatment in the p olytopal case

Let P b e a simple np olytop e with m facets F F Fix a commutative ring

k with unit Let kv v b e the p olynomial algebra over k on m generators

We make it a graded algebra by setting degv

eisner ring of a simple p oly Definition The face ring or the StanleyR

top e P is the quotient ring

I kP kv v

P m

where I is the ideal generated by all squarefree monomials v v v such that

P i i i

1 2 s

F F in P i i

i i s

1 s

Since I is a homogeneous ideal kP is a graded kalgebra

Remark In certain circumstances it is convenient to cho ose a dierent grad

ing in kv v and corresp ondingly kK These cases will b e particularly

mentioned

Example Let P b e the nsimplex regarded as a simple p olytop e

Then

kP kv v v v v

n n

Let P b e the cub e I Then

kP kv v v v v v v v v

Let P b e the mgon m > Then

I v v i j mo d m

i j P

CHAPTER

simplicial Top ology and combinatorics of

complexes

Simplicial complexes or triangulations rst intro duced by Poincare provide an

elegant rigorous and convenient to ol for studying top ological invariants by com

binatorial metho ds The algebraic top ology itself evolved from studying triangu

lations of top ological spaces With the app earance of cellular or CW complexes

algebraic to ols gradually replaced the combinatorial ones in top ology However

simplicial complexes have always played a signicant role in PL top ology discrete

and combinatorial geometry The convex geometry provides an imp ortant class of

sphere triangulations which are the b oundary complexes of simplicial p olytop es

The emergence of computers resulted in regaining the interest to Combinatorial

Top ology since simplicial complexes provide the most eective way to translate

top ological structures into machine language So it seems to b e the prop er time for

top ologists to make use of remarkable achievements in discrete and combinatorial

geometry of the last decades which we started to review in the previous chapter

simplicial complexes and p olyhedrons Abstract

Let S b e a nite set Given a subset S we denote its cardinality by j j

Definition An abstract simplicial complex on the set S is a collection

K f g of subsets of S such that for each K all subsets of including also

b elong to K A subset K is called an abstract simplex of K Oneelement

subsets are called vertices of K If K contains all oneelement subsets of S then

we say that K is a simplicial complex on the vertex set S The dimension of an

abstract simplex K is its cardinality minus one dim j j The dimension

of an abstract simplicial complex is the maximal dimension of its simplices A

simplicial complex K is pure if all its maximal simplices have the same dimension

A sub collection K K which is also a simplicial complex is called a subcomplex

of K

In most of our constructions it is safe to x an ordering in S and identify S

with the index set m f mg This makes the notation more clear however

in some cases it is more convenient to keep unordered sets

To distinguish from abstract simplices the convex p olytop es intro duced in

Example ie the convex hulls of anely indep endent p oints will b e referred

to as geometrical simplices

Definition A geometrical simplicial complex or a polyhedron is a subset

R represented as a nite union U of geometrical simplices of any dimensions P

in such a way that the following two conditions are satised

a each face of a simplex in U b elongs to U

TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

b the intersection of any two simplices in U is a face of each

A geometrical simplex from U is called a face of P as usual onedimensional faces

are vertices The dimension of P is the maximal dimension of its faces

Agreement The notion of p olyhedron from Denition is not the same as

that from Denition The rst meaning of the term p olyhedron ie the un

b ounded p olytop e is adopted in the convex geometry while the second one ie

the geometrical simplicial complex is used in the combinatorial top ology Since

b oth terms have b ecome standard in the appropriate science we cannot change their

names completely We will use p olyhedron for a geometrical simplicial complex

and convex p olyhedron for an unb ounded p olytop e Anyway it will b e always

clear from the context which p olyhedron is under consideration

In the sequel b oth abstract and geometrical simplicial complexes are assumed

to b e nite

Agreement Dep ending on the context we will denote by three dier

ent ob jects the abstract simplicial complex consisting of al l subsets of m

the convex p olytop e from Example ie the geometrical simplex and the

geometrical simplicial complex which is the union of all faces of the geometrical

simplex

Definition Given a simplicial complex K on the vertex set m say that

a p olyhedron P is a geometrical realization of K if there is a bijection b etween the

set m and the vertex set of P that takes simplices of K to vertex sets of faces

of P

If we do not care ab out the dimension of the ambient space then there is the

following quite obvious way to construct a geometrical realization for any simplicial

complex K

Construction Supp ose K is a simplicial complex on the set m Let e

denote the ith unit co ordinate vector in R For each subset m denote by

the convex hull of vectors e with i Then is a regular geometrical

simplex The p olyhedron

is a geometrical realization of K

The ab ove construction is just a geometrical interpretation of the fact that any

simplicial complex on m is a sub complex of the simplex At the same time

it is a classical result that any ndimensional abstract simplicial complex K

admits a geometrical realization in n dimensional space

Example Let S b e a simplicial np olytop e Then its b oundary S is a

geometrical simplicial complex homeomorphic to an n sphere This example

will b e imp ortant in section

Definition The f vector the hvector and the g vector of an n

dimensional simplicial complex K are dened in the same way as for simplicial

p olytop es Namely f K f f f where f is the numb er of i

n i

n n

dimensional simplices of K and h K h h h where h are

n i

determined by Here we also assume f If K S the b oundary

of a simplicial np olytop e S then one obviously has f K f S

BASIC PL TOPOLOGY AND OPERATIONS WITH SIMPLICIAL COMPLEXES

Basic PL top ology and op erations with simplicial complexes

For a detailed exp osition of PL piecewise linear top ology we refer to the

classical monographs by Hudson and by Rourke and Sanderson The

role of PL category in the mo dern top ology is describ ed for instance in the more

recent b o ok by Novikov

Definition Let K K b e simplicial complexes on the sets m m

resp ectively and P P their geometrical realizations A map m m is

said to b e a simplicial map b etween K and K if K for any K A

simplicial map is said to b e nondegenerate if j j j j for any K On

the geometrical level a simplicial map extends linearly on the faces of P to a map

P P denoted by the same letter for simplicity We refer to the latter

map as a simplicial map of polyhedrons A simplicial isomorphism of p olyhedrons

is a simplicial map for which there exists a simplicial inverse A p olyhedron P is

called a subdivision of p olyhedron P if each simplex of P is contained in a simplex

of P and each simplex of P is a union of nitely many simplices of P A PL map

P P is a map that is simplicial b etween some sub divisions of P and P

A PL homeomorphism is a PL map for which there exists a PL inverse Two PL

homeomorphic p olyhedrons sometimes are also called combinatorial ly equivalent

In other words two p olyhedrons P P are PL homeomorphic if and only if there

exists a p olyhedron P isomorphic to a sub division of each of them

Example For any simplicial complex K on m there exists a simplicial

map inclusion K

There is an obvious simplicial homeomorphism b etween any two geometrical

realizations of a given simplicial complex K This justies our single notation jK j

for any geometrical realization of K Whenever it is safe we do not distinguish

b etween abstract simplicial complexes and their geometrical realizations For ex

ample we would say simplicial complex K is PL homeomorphic to X instead of

the geometrical realization of K is PL homeomorphic to X

Construction join of simplicial complexes Let K K b e simplicial

complexes on sets S and S resp ectively The join of K and K is the simplicial

complex

K K S S K K

n n

1 2

on the set S S If K is realized in R and K in R then there is obvious

n n n n

1 2 1 2

canonical geometrical realization of K K in R R R

m m m m

1 2 1 2

Example If K K then K K

The simplicial complex K the join of K and a p oint is called the cone

over K and denoted coneK

Let S b e a pair of disjoint p oints a sphere Then S K is called the

K The geometric realization of coneK of K suspension of K and denoted

is the top ological cone susp ension over jK j

Let P and P b e simple p olytop es Then

P P P P P P

see Construction

TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

Construction The fact that the pro duct of two simplices is not a sim

plex causes some problems with triangulating the pro ducts of spaces However

there is a canonical triangulation of the pro duct of two p olyhedra for each choice

of orderings of their vertices So supp ose K K are simplicial complexes on m

and m resp ectively this is one of the few constructions where the ordering of

vertices is signicant Then we construct a new simplicial complex on m m

which we call the Cartesian product of K and K and denote K K as follows

By denition a simplex of K K is a subset of some with K

K that establishes a nondecreasing corresp ondence b etween and

More formally

K K K K

and i 6 i implies j 6 j for any two pairs i j i j

The p olyhedron jK K j denes a canonical triangulation of jK j jK j

Construction connected sum of simplicial complexes Let K K b e

two pure n dimensional simplicial complexes on sets S S resp ectively jS j

m jS j m Supp ose we are given two maximal simplices K K

Fix an identication of and and denote by S S the union of S and S

with and identied the subset created by the identication is denoted We

have jS S j m m n Both K and K now can b e viewed as collections

of subsets of S S We dene the connected sum of K at and K at to

b e the simplicial complex

K K K K n f g

1 2

on the set S S When the choices of and identication of and are

clear we use the abbreviation K K Geometrically the connected sum of jK j

and jK j at and is pro duced by attaching jK j to jK j along the faces

and then removing the face obtained from the identication of with

Example Let K b e an n simplex and K a pure n

dimensional complex with a xed maximal simplex Then K K K n f g

ie K K is obtained by deleting the simplex from K

Then K P Let P and P b e simple p olytop es Set K P

K K P P

see Construction

Definition The barycentric subdivision of an abstract simplicial com

plex K is the simplicial complex K on the set f K g of simplices of K whose

simplices are chains of emb edded simplices of K That is f g K if and

only if in K after p ossible reordering

n n

The barycenter of a p olytopal simplex R with vertices v v

n n

is the p oint b c v v The barycentric subdivision

P of a p olyhedron P is dened as follows The vertex set of P is formed by the

g barycenters of simplices of P A collection of barycenters fb c b c

spans a simplex of P if and only if in P Obviously jK j jK j

for any abstract simplicial complex K

Example For any n dimensional simplicial complex K on m

there is a nondegenerate simplicial map K dened on the vertices by

BASIC PL TOPOLOGY AND OPERATIONS WITH SIMPLICIAL COMPLEXES

j j K Here is regarded as a vertex of K and j j as a vertex of

Example Let K b e a simplicial complex on a set S and supp ose we are

given a choice function f K S assigning to each simplex K a p oint in For

instance if S m we can take f min that is assign to each simplex its minimal

vertex For every such map f there is a canonical simplicial map r K K

constructed as follows By the denition of K the vertices of K are in onetoone

corresp ondence with the simplices of K For each K regarded as a vertex of

K set r f This extends to the simplices of K by

r ff f f g

f r r

The latter is a subset of whence it is a simplex of K Thus r is indeed a

r f

simplicial map

Example order complex of a p oset Let S b e any p oset Dene ordS

to b e the collection of all chains x x x x S Then ordS is clearly

k i

a simplicial complex It is called the order complex of the p oset S The order

complex of the inclusion p oset of nonempty simplices of a simplicial complex K is

its barycentric sub division K If we add the empty simplex to the p oset then the

resulting order complex will b e cone K

Definition A simplicial complex K is called a ag complex if any set

of vertices which are pairwise connected spans a simplex of K

Proposition For each simplicial graph dimensional simplicial com

plex there exists a unique ag complex K on the same vertex set whose

skeleton is

Proof The simplices of K are the vertex sets of complete subgraphs in

Definition The minimal simplicial complex that contains a given com

plex K and is ag is called the agication of K and denoted aK

Definition Given a simplicial complex K on S a missing face of K is

a subset m such that K but every prop er subset of is a simplex of K

The following statement is straightforward

Proposition K is a ag complex if and only if every missing face has

two vertices

Example Order complexes of p osets in particular barycentric sub

divisions are examples of ag complexes On the other hand the b oundary of a

gon is ag complex but not an order complex of p oset

Let K K K see Construction Then is a missing face of

1 2

K provided that at least one of K and K is not a simplex

Definition The link and the star of a simplex K are the sub com

plexes

link K K

star K K K

TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

For any vertex v K the sub complex star v can b e identied with the cone over

link v The p olyhedron j star v j consists of all faces of jK j that contain v We

K K

omit the subscripts K whenever the context allows

For any sub complex L K dene the closed combinatorial neighborhood

U L of L in K by

U L star

K K

Equivalently the combinatorial neighb orho o d U L consists of all simplices of

K together with all their faces having some simplex of L as a face Dene also

the open combinatorial neighborhood L of jLj in jK j as the union of relative

interiors of faces of jK j having some simplex of jLj as their face

For any subset S dene the ful l subcomplex K by

K K

Set core S fv S star v K g The core of K is the sub complex core K

Thus the core is the maximal sub complex containing all vertices whose K

S core

stars do not coincide with K

Example link K

Let K b e the b oundary of the tetrahedron on four vertices

and f g Then link is the sub complex consisting of two disjoint p oints

and

Let K b e the cone over L with vertex v Then link v L star v K and

core K L

Example dual simplicial complex Let K b e a simplicial complex on

S Supp ose that K is not the full simplex on S Dene

K S S n K

Then K is also a simplicial complex on S It is called the dual of K

The dual simplicial complex K provides the following purely simplicial in

terpretation for the Alexander duality see eg p b etween jK j and

S n jK j for any simplicial complex K emb edded in the m sphere Let

us consider the barycentric sub division of the b oundary of a geomet

rical simplex on the vertex set m f mg By the denition the faces

of corresp ond to the pairs of subsets of m satisfying j j >

j j 6 m Denote the corresp onding faces by For example is

figfig

m m

the vertex v fig of regarded as a vertex of Denote i m n fig

and more generally b m n for any subset m For any simplicial complex

K on m dene the following sub complex in

D K

b b

Proposition For any simplicial complex K on the set m the

polyhedron D K provides a geometrical realization for the barycentric subdivision

of the dual simplicial complex

D K jK j

BASIC PL TOPOLOGY AND OPERATIONS WITH SIMPLICIAL COMPLEXES

Moreover if the barycentric subdivision of K is realized canonical ly as a subpoly

hedron in then

m1 0

jK j n jK j

Proof The complete pro of is elementary but quite tedious We just give an

illustrating picture Figure Here K is the b oundary of the square on vertices

Then K consists of two disjoint segments The picture shows b oth K

and K as sub complexes in

jK j

b b

d d

t t

Z Z Z Z Z Z Z Z Z

Z Z Z Z Z Z Z Z Z t

Figure Dual complex and Alexander duality

Corollary Simplicial Alexander duality For any simplicial complex

K on the set m it holds that

e b e

H K H K 6 j 6 m

e e

where H and H denotes the k th reduced simplicial homology and cohomology

groups with integer coecients respectively We use the agreement H

H Z here

m m

Proof Since is homeomorphic to S the Alexander duality the

orem and Prop osition show that

e b e

m1 0

jK j H K H D n

j j

mj m

e e

H H K 6 j 6 m S n K

Corollary admits the following generalization which we will use in Chap

ter

TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

Proposition For any simplicial complex K on m and simplex

K it holds that

mj j j

e e

H H link K

b b

where b m n and K is the ful l subcomplex in K dened in

Corollary is obtained by substituting ab ove

Example Let K b e the b oundary of a p entagon Then K is the Mobius

b b b b

band triangulated as it is shown on Figure If we map the p oints to the

vertices of a simplex and to its barycenter then the whole triangulated Mobius

band K b ecomes a sub complex in the dimensional Schlegel diagram see

Lecture of a dimensional simplex

b b b

r r r

A A A

Q r r

A A A

A A r Ar r r

b b b b

Cr r

K K

Figure The b oundary of p entagon and its dual

Simplicial spheres

Definition A simplicial q sphere is a simplicial complex K homeomor

phic to q sphere A PL sphere is a simplicial sphere K which is PL homeomorphic

to the b oundary of a simplex equivalently there is a sub division of K isomorphic

to a sub division of the b oundary of A homology q sphere is a top ological

manifold which has the same homology as the q sphere S

The b oundary of a simplicial np olytop e is an n dimensional PL sphere

A PL sphere simplicially isomorphic to the b oundary of a simplicial p olytop e is

called a polytopal sphere We have the following hierarchy of combinatorial ob jects

p olytopal spheres PL spheres simplicial spheres

In dimension any simplicial sphere is p olytopal see eg or Theo

rem However in higher dimensions b oth ab ove inclusions are strict The

rst inclusion in is strict already in dimension Namely there are com

binatorially dierent triangulations of the sphere with vertices out of which

are nonpolytopal The rst one now known as the Br uckner sphere was found by

Grun baum x see also as a correction of Bruc kners result of

on the classication of simplicial p olytop es with vertices The second known

as Barnette sphere is describ ed in The complete classication of simplicial

spheres with up to vertices was obtained in We mention also the result

of Mani that any simplicial q sphere with up to q vertices is p olytopal

TRIANGULATED MANIFOLDS

As for the second inclusion in it is known that in dimension any simplicial

sphere is PL In dimension the corresp onding question is op en see the discussion

in the next section but starting from dimension there exist nonPL simplicial

spheres One such thing is describ ed in Example b elow According to the

result of for any n > there is a nonPL triangulation of S with n

vertices

Since the f vector of a p olytopal sphere coincides with the f vector of the cor

resp onding simplicial p olytop e see Denition the g theorem Theorem

holds for p olytopal spheres So it is natural to ask whether the g theorem extends

to simplicial spheres This question was p osed by McMullen as an extension

of his conjecture for simplicial p olytop es Since when McMullens conjec

ture for simplicial p olytop es was proved by Billera Lee and Stanley the following

is regarded as the main op en combinatorialgeometrical problem concerning the

f vectors of simplicial complexes

Problem g conjecture for simplicial spheres Does the g theorem The

orem hold for simplicial spheres

The g conjecture is op en even for PL spheres Note that only the necessity

of g theorem ie that the g vector is an M vector is to b e veried for simplicial

spheres If correct the g conjecture would imply a characterisation of f vectors of

simplicial spheres

The rst part of Theorem the DehnSommerville equations is known to

b e true for simplicial spheres see Corollary b elow Simplicial spheres also

satisfy the UBC and the LBC inequalities as stated in Theorems and The

LBC in particular the inequality h 6 h for spheres was proved by Barnette

see also The UBC for spheres is due to Stanley see Corollary

b elow This implies that the g conjecture is true for simplicial spheres of dimen

sion 6 The inequality h 6 h from the GLBC is op en Many attempts to

prove the g conjecture were made during the last two decades Though unsuccessful

these attempts resulted in some very interesting reformulations of the g conjecture

The results of Pachner reduce the g conjecture for PLspheres to

some prop erties of bistel lar moves see the discussion after Theorem We also

mention the results of showing that the g conjecture follows from the skeletal

r rigidity of simplicial n sphere for r 6 It was shown indep endently by

Kalai and Stanley Corollary that the GLBC holds for the b oundary of

an ndimensional ball that is a sub complex of the b oundary complex of a simplicial

n p olytop e However it is not clear now which simplicial complexes o ccur in

this way The lack of progress in proving the g conjecture motivated Bjorner and

Lutz to launch a computeraided search for counterexamples Though their bis

tellar ip algorithm and computer program BISTELLAR pro duced many remark

able results on triangulations of manifolds no counterexamples to the g conjecture

were found For more history of g theorem and related questions see

Lecture

Triangulated manifolds

Definition A simplicial complex K is called a triangulated manifold or

simplicial manifold if the p olyhedron jK j is a top ological manifold All manifolds

considered here are compact connected and closed unless otherwise stated A

q dimensional PL manifold or combinatorial manifold is a simplicial complex K

TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

such that link is a PL sphere of dimension q j j for every nonempty simplex

Every PL manifold K is a triangulated manifold Indeed for each vertex

v K the q dimensional PLsphere link v b ounds an op en neighb orho o d U

which is homeomorphic to an op en q ball Since any p oint of jK j is contained in

U for some v this denes an atlas for jK j

Do es every triangulation of a top ological manifold yield a simplicial complex

which is a PL manifold The answer is no and the question itself ascends to

a famous conjecture of the dawn of top ology known as the Hauptvermutung der

Topologie In the early days of top ology all of the known top ological invariants were

dened in combinatorial terms and it was very imp ortant to nd out whether the

top ology of a p olyhedron fully determines the combinatorics of triangulation In

the mo dern terminology the Hauptvermutung conjecture states that any two home

omorphic p olyhedrons are combinatorially equivalent PL homeomorphic This is

valid in dimensions 6 the result is due to Rado for manifolds Papakyri

akop oulos for complexes Moise for manifolds and E Brown

for complexes see for the mo dern exp osition The rst examples of com

plexes disproving the Hauptvermutung in dimensions > were found by Milnor in

the early s However the manifold Hauptvermutung namely the question of

whether two homeomorphic triangulated manifolds are combinatorially equivalent

had remained op en until the s It was nally disproved with the app earance

of the following theorem

of any ho Theorem Edwards Cannon The double suspension S

n n

is homeomorphic to S mology nsphere S

This theorem was proved by Edwards for some particular homology

spheres and by Cannon in the general case The following example pro

vides a nonPL triangulation of the sphere and therefore disproves the manifold

Hauptvermutung in dimensions >

b e any simplicial homol Example nonPL simplicial sphere Let S

ogy sphere which is not a top ological sphere The Poinc are sphere SO A tri

angulated in any way provides an example of such a manifold By Theorem

is homeomorphic to S and more generally S the double susp ension S

H H

cannot b e PL since S is homeomorphic to S for k > However S

H H

app ears as the link of some simplex in S

In the p ositive direction it is known that two homeomorphic simply connected

PL manifolds of dimension > with no torsion in third homology group are combi

natorially equivalent PL homeomorphic This is Sullivans famous Hauptvermu

tung theorem The general classication of PL structures on higher dimensional

top ological manifolds was obtained by Kirby and Sieb enmann see

The following theorem gives a characterization of simplicial complexes which

are triangulated manifolds of dimension > and generalizes Theorem

Theorem Edwards For q > the polyhedron of a simplicial com

plex K is a topological q manifold if and only if link has the homology of a

q j jsphere for each nonempty simplex j j K and link v is simply connected

for each vertex v K

BISTELLAR MOVES

The discovery of nonPL triangulations of top ological manifolds motivated fur

ther questions Among them is whether every top ological manifold admits a PL

triangulation or at least any triangulation not necessarily PL Another related

question is whether the Hauptvermutung is valid in dimension Both questions

were answered negatively by the results of Freedman and Donaldson early s

A smo oth manifold can b e triangulated by Whitneys theorem All top olog

ical and dimensional manifolds can b e triangulated as well for manifolds

see Moreover since the link of a vertex in a simplicial sphere is a sphere

and a sphere is always PL all and manifolds are PL However in di

mension there exist top ological manifolds that do not admit a PLtriangulation

An example is provided by Freedmans fake C P x x a top ological

manifold which is homeomorphic but not dieomorphic to the complex pro jective

plane C P This shows that the Hauptvermutung is false for dimensional man

ifolds Even worse as it is shown in there exist top ological manifolds eg

Freedmans top ological manifold with the intersection form E that do not admit

any triangulation In dimensions > the triangulation problem is op en

Problem Triangulation Conjecture Is it true that any topological man

ifold of dimension > can be triangulated

Another wellknown problem of PLtop ology concerns the uniqueness of a PL

structure on a top ological sphere

Problem Is a PL manifold homeomorphic to the topological sphere

necessarily a PL sphere

Four is the only dimension where the uniqueness of a PL structure on a top olog

ical sphere is op en For dimensions 6 the uniqueness was proved by Moise

and for dimensions > it follows from the result of Kirby and Sieb enmann

In dimension the category of PL manifolds is equivalent to the smo oth category

hence the ab ove problem is equivalent to if there exists an exotic or fake sphere

The history of the Hauptvermutung conjecture is summarized in a survey ar

ticle by A Ranicki This source also contains a more detailed discussion

of recent developments and op en problems including those mentioned ab ove in

combinatorial and PL top ology

Bistellar moves

Bistellar moves in other notation bistellar ips or bistellar op erations were

intro duced by Pachner see as a generalization of stel lar subdivisions

These op erations allow us to decomp ose a PL homeomorphism into a sequence

of simple ips and thus provide a very convenient way to compute and handle

top ological invariants of PL manifolds Starting from a given PL triangulation

bistellar op erations may b e used to construct new triangulations with some go o d

prop erties eg symmetric or with a small numb er of vertices On the other hand

bistellar moves can b e used to pro duce some nasty triangulation if we start from a

nonPL one Both approaches were applied in to nd many interesting triangu

lations of lowdimensional manifolds Bistellar moves also provide a combinatorial

interpretation for algebraic blow up and blow down op erations for pro jective toric

varieties see section as well as for some top ological surgery operations see

Construction Finally bistellar moves may b e used to dene a metric on the

space of PL triangulations of a given PL manifold see for more details

TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

Definition Let K b e a simplicial q manifold or any pure q dimensional

simplicial complex and K a q isimplex 6 i 6 q such that link is

the b oundary of an isimplex that is not a face of K Then the op eration

on K dened by

K K n

is called a bistel lar imove Bistellar imoves with i > are also called reverse

q imoves Note that a move adds a new vertex to a triangulation we assume

that D a reverse move deletes a vertex while all other bistellar moves

do not change the numb er of vertices see Figures and Two pure simplicial

complexes are bistel larly equivalent if one is taken to another by a nite sequence

of bistellar moves

S S

move

S S

move

PS S

move

Figure Bistellar moves for q

Remark The bistellar move is just the stellar sub division or connected sum

with the b oundary of a simplex In particular stacked spheres ie the b ound

aries of stacked p olytop es see Denition are exactly those obtained from the

b oundary of a simplex by applying bistellar moves

It is easy to see that two bistellarly equivalent PL manifolds are PL homeo

morphic The following remarkable result shows that the converse is also true

Theorem Pachner Theorem Two PL manifolds

are bistel larly equivalent if and only if they are PL homeomorphic

The b ehavior of the face numb ers of a triangulation under bistellar moves is

easily controlled Namely the following statement holds

Theorem Pachner Let L be a q dimensional triangulated mani

fold obtained from K by applying a bistel lar k move 6 k 6 Then

g L g K

k k

g l g K for al l i k

i i

where g K h K h K i 6 are the components of the g vector

i i i

Moreover if q is even and k then g L g K for al l i

i i

BISTELLAR MOVES

move

Q Q

Q move Q

Q Q

move

H H

move

H H

B B

Figure Bistellar moves for q

This theorem allows us to interpret the inequalities from the g conjecture for

PL spheres see Theorem in terms of the numb ers of bistellar k moves needed

to transform the given PL sphere to the b oundary of a simplex For instance the

inequality h 6 h n > is equivalent to the statement that the numb er of

moves in the sequence of bistellar moves taking a given n dimensional PL

sphere to the b oundary of an nsimplex is lesser than or equal to the numb er of

reverse moves Note that the g vector of has the form

Remark Pachner also proved an analogue of Theorem for PL manifolds

with boundary see For this purp ose he intro duced another class of

op erations on triangulations called elementary shel lings

TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

CHAPTER

e and homological algebra of Commutativ

simplicial complexes

The app earance of the StanleyReisner face ring of simplicial complex at the b egin

ning of s outlined a new approach to combinatorial problems concerning sim

plicial complexes It relies up on the interpretation of combinatorial prop erties of

simplicial complexes as algebraic prop erties of the corresp onding face rings and uses

commutative algebra machinery such as CohenMacaulay and Gorenstein algebras

lo cal cohomology etc The main reference here is R Stanleys monograph

StanleyReisner face rings of simplicial complexes

Recall that kv v denotes the graded p olynomial algebra over a com

mutative ring k with unit deg v

Definition The face ring or the StanleyReisner ring of a simplicial

complex K on the vertex set m is the quotient ring

kK kv v I

m K

where I is the homogeneous ideal generated by all squarefree monomials v

v v v i i such that fi i g is not a simplex of K The

i i i s s

1 2 s

ideal I is called the StanleyReisner ideal of K

Supp ose P is a simple np olytop e P its p olar and K the b oundary of P

Then K is a p olytopal simplicial n sphere The face ring of P from Deni

tion coincides with that of K from the ab ove denition kP kK

P P

Example Let K b e a dimensional simplicial complex shown on Fig

ure Then

I v v v v v v v v v v

B r r

H r r

Figure

COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

The StanleyReisner ring kK is a quadratic algebra ie the ideal I is

generated by quadratic monomials if and only if K is a ag complex see Deni

tion and compare with Prop osition

Let K K b e the join of K and K see Construction Then

kK K kK kK

In particular for any two simple p olytop es P and P we have

kP P kP kP

see Construction

Let K K b e the connected sum of two pure n dimensional

1 2

simplicial complexes on sets S S resp ectively regarded as a simplicial complex on

is generated the set S S see Construction Then the ideal I

K K

2 1

1 2

by the ideals I I and the monomials v and v v where i S n and

K K i i

1 2 1 2

i S n

For any subset fi i g m denote by v the squarefree monomial

v v Note that the ideal I is monomial and has basis of monomials v

i i K

corresp onding to missing faces of K

Proposition Every squarefree monomial ideal in the polynomial ring

has the form I for some simplicial complex K

Proof Let I b e a squarefree monomial ideal Set

K f m v I g

Then one easily checks that K is a simplicial complex and I I

Proposition Let K K be a simplicial map see Denition

between two simplicial complexes K and K on the vertex sets m and m

respectively Dene the map kw w kv v by

m m

2 1

w v

j i

figf fj g

Then descends to a homomorphism kK kK which we wil l also denote

Proof We have to check that I I Supp ose fj j g

K K s

2 1

m is not a simplex of K Then

w w v v

j j i i

1 s 1 s

1 1

fi g fi g fj g fj g

1 s s 1

We claim that fi i g is not a simplex of K for any monomial v v

s i i

1 s

in the right hand side of the ab ove identity Indeed otherwise we would have

K by the denition of simplicial map which is imp ossible Hence the

right hand side of is in I

Example The face ring of the barycentric sub division K of K is

kK kb K I

STANLEYREISNER FACE RINGS OF SIMPLICIAL COMPLEXES

where b is the p olynomial generator corresp onding to simplex K We have

the simplicial map r K K see Example Then it is easy to see that

r v b

K min j

for any generator v kK

Example The nondegenerate map K from Example in

duces the following map of the corresp onding StanleyReisner rings

kv v kK

v b

j ji

This denes a canonical kv v mo dule structure in kK

Definition Let M M M b e a graded kmo dule The series

i i

dim M F M t t

is called the Poincare series of M

Remark In the algebraic literature the series F M t is called the Hilbert

series or HilbertPoincare series

The following lemma may b e considered as an algebraic denition of the h

vector of a simplicial complex

Lemma Stanley Theorem I I The Poincare series of kK

can be calculated as

n i

h h t h t f t

n i

F kK t

i n

t t

where f f is the f vector and h h is the hvector of K

n n

k +1

Proof Any monomial in kK has the form v v where

i i

k +1

fi i g is a simplex of K and are some p ositive integers

k k

2(k +1)

to the Poincare Thus every k simplex of K contributes the summand

2 k +1

series which proves the rst identity The second identity is an obvious corollary

Example Let K Then f for 6 i 6 n h

and h for i Since any subset of n is a simplex of we have

n n n

k kv v and F k t t which agrees with

Lemma

Let K b e the b oundary of an nsimplex Then h i n and

kK kv v v v v By Lemma

n n

t t

F kK t

COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

CohenMacaulay rings and complexes

Here we supp ose k is a eld Let A b e a nitelygenerated commutative graded

algebra over k We also assume that A has only evendegree graded comp onents

so it is commutative in b oth the usual and the graded sense

l dimension of A denoted Kd A is the maxi Definition The Krul

mal numb er of algebraically indep endent elements of A A sequence

of n Kd A homogeneous elements of A is called an hsop homogeneous system of

parameters if the Krull dimension of the quotient A is zero Equiva

lently is an hsop if n Kd A and A is a nitelygenerated k

n n

mo dule The elements of an hsop are algebraically indep endent

Lemma No ether normalization lemma For any nitelygenerated gra

ded algebra A there exists an hsop If k is of zero characteristic and A is generated

by degreetwo elements then a degreetwo hsop can be chosen

In the case when A is generated by degreetwo elements a degreetwo hsop is

called an lsop linear system of parameters

Remark If k is of nite characteristic then an lsop may fail to exist for alge

bras generated in degree two see Example b elow

In the rest of this chapter we assume that the eld k is of zero characteristic

Definition A sequence of homogeneous elements of A is

called a regular sequence if is not a zero divisor in A for 6 i k

i i

ie the multiplication by is a monomorphism of A into itself

i i

Equivalently is a regular sequence if are algebraically inde

k k

p endent and A is a free k mo dule

Remark The concept of a regular sequence can b e extended to nonnitely

generated graded algebras and to algebras over any integral domain Regular se

quences in graded p olynomial rings R a a on innitely many generators

where deg a i and R is a subring of the eld Q of rationals are used in the

algebraic top ology for constructing complex cob ordism theories with co ecients

see

Any two maximal regular sequences have the same length which is called the

depth of A and denoted depth A Obviously depth A 6 Kd A

Definition Algebra A is called CohenMacaulay if it admits a regular

sequence of length n Kd A

A regular sequence of length n Kd A is an hsop It follows that

A is CohenMacaulay if and only if there exists an hsop such that A is

a free k mo dule If in addition A is generated by degreetwo elements

then one can cho ose to b e an lsop In this case the following formula for

the Poincare series of A holds

F A t

where F A t h h t is a p olynomial The nite vector

h h is called the hvector of A

COHENMACAULAY RINGS AND COMPLEXES

Definition A simplicial complex K is called CohenMacaulay over

k if its face ring kK is CohenMacaulay

Obviously Kd kK n Lemma shows that the hvector of kK

coincides with the hvector of K

Example Let K b e the b oundary of a simplex Then kK

kv v v v v v The elements v v kK are algebraically indep endent

but do not form an hsop since kK v v kv and Kd kK v v

On the other hand the elements v v v v of kK form an hsop

since kK ktt It is easy to see that kK is a free k mo dule

with one dimensional generator one dimensional generator v and one

dimensional generator v Thus kK is CohenMacaulay and is a regular

sequence

Theorem Stanley If K is a CohenMacaulay simplicial complex

then hK h h is an M vector see Denition

Proof Let b e a regular sequence of degreetwo elements of kK

Then A kK is a graded algebra generated by degreetwo elements

and dim A h Now the result follows from Theorem

k i

The following fundamental theorem characterizes CohenMacaulay complexes

combinatorially

Theorem Reisner A simplicial complex K is CohenMacaulay

over k if and only if for any simplex K including and i dimlink

e e

H link k Here H X k denotes the ith reduced homology group of X

i i

with coecients in k

Corollary A simplicial sphere is a CohenMacaulay complex

Theorem shows that the hvector of a simplicial sphere is an M vector

This argument was used by Stanley to extend the UBC Theorem to simplicial

spheres

Corollary Upp er Bound Theorem for spheres Stanley The h

vector h h h of a simplicial n sphere K with m vertices satises

mni

h K 6 6 i

Hence the UBC holds for simplicial spheres that is

n n

f K 6 f C m for i n

i i

see Corol lary

Proof Since h K is an M vector there exists a graded algebra A

n i

A A A generated by degreetwo elements such that dim A h

k i

Theorem In particular dim A h m n Since A is generated by A

the numb er h cannot exceed the total numb er of monomials of degree i in m n

mni

variables The latter is exactly i

COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

Homological algebra background

Here we review some homological algebra Unless otherwise stated all mo dules

in this section are assumed to b e nitelygenerated graded kv v mo dules

deg v

Definition A nite free resolution of a mo dule M is an exact sequence

d d d d

h h

R R R R M

are degree where the R are nitelygenerated free mo dules and the maps d

preserving The minimal numb er h for which a free resolution exists is called

the homological dimension of M and denoted hd M By the Hilb ert syzygy theorem

a nite free resolution exists and hd M 6 m A resolution can b e written

ij ij i j

as a free bigraded dierential module R d where R R R R

the j th graded comp onent of the free mo dule R The cohomology of R d

is zero in nonzero dimensions and H R d M Conversely a free bigraded

ij ij ij

dierential mo dule R R d R R with H R d M

ij >

i i i

and H R d for i denes a free resolution with R R

Remark For the reasons sp ecied b elow we numerate the terms of a free

resolution by nonpositive numb ers thereby viewing it as a cochain complex

The Poincare series of M can b e calculated from any free resolution by

means of the following classical theorem

Theorem Suppose that R has rank q with free generators in degrees

d d i h Then

i q i

m i d d

1i q i

F M t t t t

Proof By the denition of resolution the following map of co chain complexes

d d d d

h h

R R R R

y y y y

is a quasiisomorphism ie induces an isomorphism in the cohomology Equating

the Euler characteristics of b oth complexes in each degree we get

Construction There is the following straightforward way to construct

a free resolution for a mo dule M Take a set of generators a a for M and

dene R to b e a free kv v mo dule with k generators in the corresp onding

degrees There is an obvious epimorphism R M Then take a set of generators

a a in the kernel of R M and dene R to b e a free kv v

k m

mo dule with k generators in the corresp onding degrees and so on On the ith

step we take a set of generators in the kernel of the previously constructed map

i i i

d R R and dene R to b e a free mo dule with the corresp onding

generators The Hilb ert syzygy theorem guarantees this pro cess to end up at most

at the mth step

HOMOLOGICAL ALGEBRA BACKGROUND

Example minimal resolution For graded nitely generated mo dules

M a minimal generator set or a minimal basis can b e chosen This is done

as follows Take the lowest degree in which M is nonzero and there cho ose a vec

tor space basis Span a mo dule M by this basis and then take the lowest degree

in which M M In this degree cho ose a vector space basis in the complement

of M and span a mo dule M by this basis and M Then continue this pro cess

Since M is nitely generated on some pth step we get M M and a basis for M

with a minimal numb er of generators

If we take a minimal set of generators for mo dules at each step of Construc

tion then the pro duced resolution is called minimal Each of its terms

R has the smallest p ossible rank see Example b elow There is also the

following more formal but less convenient for particular computations deni

tion of minimal resolution see Let M M b e two mo dules Set J M

v M v M v M M A map f M M is called minimal if

Ker f J M A resolution is called minimal if all maps d are minimal

A minimal resolution is unique up to an isomorphism

Example Koszul resolution Let M k The kv v mo dule

structure on k is dened via the map kv v k that sends each v to

m i

u u denote the exterior algebra on m generators Turn the tensor Let

pro duct R u u kv v here and b elow we use for into

m m k

a dierential bigraded algebra by setting

bideg u bideg v

i i

du v dv

i i i

and requiring that d b e a derivation of algebras An explicit construction of co chain

homotopy x shows that H R d for i and H R d k Since

u u kv v is a free kv v mo dule it determines a free

m m m

resolution of k This resolution is known as the Koszul resolution Its expanded

form is as follows

u u kv v

m m

u u kv v kv v k

m m m

where u u is the submo dule of u u spanned by monomials

m m

i i

of length i Thus in the notation of we have R u u

kv v

b e another mo dule then applying the functor N to we Let N

kv v

1 m

obtain the following co chain complex of graded mo dules

R N R N

kv v kv v

1 m 1 m

and the corresp onding bigraded dierential mo dule R N d The ith co

homology mo dule of the ab ove co chain complex is denoted Tor M N

kv v

1 m

Tor M N H R N d

kv v

1 m

kv v

1 m

i i

Kerd R N R N

kv v kv v

1 m 1 m

dR N

kv v

1 m

COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

Since b oth the R s and N are graded mo dules we actually have

Tor M N Tor M N

kv v

1 m

where

i j i j

Ker d R N R N

kv v kv v

1 m 1 m

Tor M N

kv v

i j

1 m

dR N

kv v

1 m

The ab ove mo dules combine to a bigraded kv v mo dule

Tor M N Tor M N

kv v

1 m

kv v

1 m

The following prop erties of Tor M N are well known see eg

kv v

1 m

Proposition a The module Tor M N does not depend up

kv v

1 m

to isomorphism on a choice of resolution

i i

b Both Tor N and Tor M are covariant functors

kv v kv v

1 m 1 m

M N M N c Tor

kv v

1 m

i i

N M M N Tor d Tor

kv v kv v

1 m 1 m

M N are dened for algebras A far In homological algebra Amo dules Tor

more general than p olynomial rings and nitelygeneratedness assumption for

mo dules M and N may b e also dropp ed Although a nite Afree resolution

of M may fail to exist in general there is always a projective resolution which allows

one to dene Tor M N in the same way as ab ove Note that pro jective mo dules

over the p olynomial algebra are free In the nongraded case this was known as the

Serre problem now solved by Quillen and Suslin However the graded version of

this fact is much easier to prove In this text the Tormo dules Tor M N over

algebras dierent from the p olynomial ring app ear only in sections and

Homological prop erties of face rings Toralgebras and Betti

numb ers

Here we apply general constructions from the previous section in the case when

M kK and N k As usual K K is assumed to b e a simplicial complex

on m Since deg v we have

kK k Tor kK k Tor

kv v

1 m

kv v

1 m

ie Tor kK k is nonzero only in even second degrees Dene the

kv v

1 m

bigraded Betti numbers of kK by

kK dim Tor kK k 6 i j 6 m

kv v

1 m

Supp ose that is a minimal free resolution of M kK Example Then

R kv v is a free kv v mo dule with one generator of degree

m m

The basis of R is a minimal generator set for I Kerkv v kK

K m

and is represented by the missing faces of K For each missing face fi i g of K

denote by v the corresp onding basis element of R Then deg v k

i i i i

1 1

k k

HOMOLOGICAL PROPERTIES OF FACE RINGS

and the map d R R takes v to v v Since the maps d in

i i i i

1 1

k k

are minimal the dierentials in the co chain complex

R k R k

kv v kv v

1 m 1 m

are trivial Hence for the minimal resolution of kK it holds that

Tor kK k R k

kv v

1 m

kv v

1 m

ij ij

kK rank R

Example Let K b e the b oundary of a square Then

kK kv v v v v v

Let us construct a minimal resolution of kK using Construction The

mo dule R has one generator of degree and the map R kK is the

quotient pro jection Its kernel is the ideal I and the minimal basis consists of

two monomials v v and v v Hence R has two free generators of degree

denoted v and v and the map d R R sends v to v v and v to

v v The minimal basis for the kernel of R R consists of one element

v v v v v v Hence R has one generator of degree say a and the map

d R R is injective and sends a to v v v v v v Thus we have the

minimal resolution

R R R M

where rank R kK rank R kK rank R

The Betti numb ers kK are imp ortant combinatorial invariants of sim

plicial complex K see The following theorem which was proved by combina

torial metho ds reduces the calculation of kK to calculating the homology

groups of sub complexes of K

Theorem Ho chster or Theorem We have

kK dim H K

k j i

m j jj

where K is the ful l subcomplex of K corresponding to see We assume

H k above

Example Again let K b e the b oundary of a square so m This

time we calculate the Betti numb ers kK using Ho chsters theorem Among

twoelement subsets of m there are four simplices and two nonsimplices namely

f g and f g Simplices contribute trivially to the sum for kK while

each of the two nonsimplices contributes hence kK Further

each of the four full sub complexes corresp onding to threeelement subsets of m is

contractible hence its reduced homology vanishes and kK for any i

Finally the full sub complex K with j j is K itself hence kK

dim H K The latter equals for i and zero otherwise

k i

In chapter we show that kK equals the corresp onding bigraded Betti

numb er of the momentangle complex Z asso ciated to simplicial complex K This

provides an alternative top ological way for calculating the numb ers kK

Now we turn to the Koszul resolution Example

COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

Lemma For any module M it holds that

Tor M k H u u M d

kv v

1 m

where H u u M d is the cohomology of the bigraded dierential module

u u M and d is dened as in

Proof Using the Koszul resolution u u kv v d in the

m m

denition of Tor k M we calculate

kv v

1 m

Tor M k Tor k M

kv v kv v

1 m 1 m

H u u M H u u kv v M

m m m

kv v

1 m

Corollary Suppose that a kv v module M is an algebra then

Tor M k is canonical ly a nitedimensional bigraded kalgebra

kv v

1 m

Proof It is easy to see that in this case the tensor pro duct u u M

is a dierential algebra and Tor M k is its cohomology by Lemma

kv v

1 m

Definition The bigraded algebra Tor M k is called the Tor

kv v

1 m

algebra of algebra M If M kK then it is called the Toralgebra of simplicial

complex K

Remark For general N k the mo dule Tor M N has no canonical

kv v

1 m

multiplicative structure even if b oth M and N are algebras

Lemma A simplicial map K K between two simplicial complexes

on the vertex sets m and m respectively induces a homomorphism

Tor kK k Tor kK k

kw w kv v

1 m 1 m

2 1

of the corresponding Toralgebras

Proof This follows directly from Prop ositions and b

Construction multigraded structure in the Toralgebra We may in

vest the p olynomial ring kv v with a multigrading more precisely N

grading by setting mdeg v where stands at the ith

v is i i Supp ose place Then the multidegree of monomial v

that algebra M is a quotient of the p olynomial ring by a monomial ideal Then the

multigraded structure descends to M and to the terms of resolution We may

assume that the dierentials in the resolution preserve the multidegrees Then the

mo dule Tor M N acquires a canonical N N grading ie

kv v

1 m

Tor M k Tor M k

kv v

1 m

kv v

1 m

i> j N

In particular the Toralgebra of K is canonically an N N graded algebra

Remark According to our agreement the rst degree in the Toralgebra is

nonpositive Rememb er that we numerated the terms of kv v free Koszul

resolution of k by nonp ositive integers In these notations the Koszul complex

M u u d b ecomes a cochain complex and Tor M k is its

kv v

1 m

cohomology not the homology as usually regarded One of the reasons for such an

HOMOLOGICAL PROPERTIES OF FACE RINGS

agreement is that Tor kK k is a contravariant functor from the cate

kv v

1 m

gory of simplicial complexes and simplicial maps see Lemma It also explains

kv v

1 m

our notation Tor M k used instead of the usual Tor M k

kv v

1 m

These notations are convenient for working with Eilenb ergMo ore sp ectral se

quences see section

The upp er b ound hd M 6 m from the Hilb ert syzygy theorem can b e replaced

by the following sharp er result

Theorem Auslander and Buchsbaum hd M m depth M

n n

In particular if M kK and K is CohenMacaulay see Deni

tion then hd kK m n and Tor kK k for i m n

kv v

1 m

From now on we assume that M is generated by degreetwo elements and

the kv v mo dule structure in M is dened through an epimorphism p

kv v M b oth assumptions are satised by denition for M kK

Supp ose that is a regular sequence of degreetwo elements of M Let J

M b e the ideal generated by Cho ose degreetwo elements

k k

t kv v such that pt i k The ideal in kv v

i m i i m

generated by t t will b e also denoted by J Then we have kv v J

k m

kw w Under these assumptions we have the following reduction lemma

Lemma The fol lowing isomorphism of algebras holds for any ideal J

generated by a regular sequence

Tor M k Tor M J k

kv v kv v J

1 m 1 m

In order to prove the lemma we need the following fact from homological alge

bra

Theorem p Let be an algebra its subalgebra and

the quotient algebra Suppose that is a free module and we are given an

module A and a module C Then there exists a spectral sequence fE d g such

r r

that

Tor A C E Tor E A Tor C k

Proof of Lemma Set kv v kt t A k

m k

C M Then is a free mo dule and kv v J Therefore

Theorem gives a sp ectral sequence

E Tor M k E Tor Tor M k k

kv v

1 m

Since is a regular sequence M is a free mo dule Therefore

Tor M k M k M J and Tor M k for q

It follows that E for q Thus the sp ectral sequence collapses at the E

term and we have

Tor M k Tor Tor M k k Tor M J k

kv v kv v J

1 m 1 m

which concludes the pro of

COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

Gorenstein complexes and DehnSommerville equations

It follows from Theorem that if M is CohenMacaulay of Krull dimen

sion n then depth M n hd M m n and Tor M k for

kv v

1 m

i m n

Definition Supp ose M is a CohenMacaulay algebra of Krull dimen

sion n Then M is called a Gorenstein algebra if Tor M k k

kv v

1 m

Following Stanley we call a simplicial complex K Gorenstein if kK is

a Gorenstein algebra Further K is called Gorenstein if kK is Gorenstein and

K core K see Denition The following theorem characterizes Gorenstein

simplicial complexes

Theorem xI I A simplicial complex K is Gorenstein over k

if and only if for any simplex K including the subcomplex link has

the homology of a sphere of dimension dim link

In particular simplicial spheres and simplicial homology spheres triangulated

manifolds with the homology of a sphere are Gorenstein complexes However the

Gorenstein prop erty do es not guarantee a complex to b e a triangulated manifold

links of vertices are not necessarily simply connected compare with Theorem

The Poincare series of the Toralgebra and the face ring of a Gorenstein complex

are self dual in the following sense

Theorem xI I Suppose K is a Gorenstein complex Then

kK k 6 the fol lowing identities hold for the Poincare series of Tor

kv v

1 m

i 6 m n

mni

kK k t kK k F Tor t F Tor

kv v

kv v t

1 m

Equivalently

mnimj ij

kK 6 i 6 m n 6 j 6 m kK

Corollary If K is Gorenstein then

F kK t F kK

Proof We apply Theorem to a minimal resolution of kK It follows

from that the numerators of the summands in the right hand side of are

kK k t i m n Hence exactly F Tor

kv v

1 m

m i

kK k t F kK t t F Tor

kv v

1 m

Using Theorem we calculate

mni

i m m

t F Tor F kK t t kK k

kv v

1 m

mnj m

F Tor kK k

t t kv v

1 m

F kK t

GORENSTEIN COMPLEXES AND DEHNSOMMERVILLE EQUATIONS

Corollary The DehnSommervil le relations h h 6 i 6 n

i ni

hold for any Gorenstein complex K in particular for any simplicial sphere

Proof This follows from Lemma and Corollary

As it was p ointed out by Stanley in Gorenstein complexes are the

most appropriate candidates for generalizing the g theorem to As we have seen

p olytopal spheres PL spheres simplicial spheres and simplicial homology spheres

are examples of Gorenstein complexes

The DehnSommerville equations can b e generalized even b eyond Gorenstein

complexes In Klee reproved the f vector version of the DehnSommer

ville equations in the more general context of Eulerian complexes A pure simplicial

complex K is called Eulerian if for any simplex K including holds

nj j nj j

link S Generalizations of equations

were obtained by Bayer and Billera for Eulerian posets and Chen and Yan

for arbitrary p olyhedra

In section we deduce the generalized DehnSommerville equations for tri

angulated manifolds as a consequence of the bigraded Poincare duality for moment

angle complexes In particular this gives the following short form of the equations

in terms of the hvector

n n i

K S i n h h

ni i

n n n

Here K f f f h is the Euler char

n n

n n n

acteristic of K and S is that of a sphere Note that the

ab ove equations reduce to the classical h h in the case when K is a simplicial

ni i

sphere or has o dd dimension

COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

CHAPTER

Cubical complexes

At some stage of development of the combinatorial top ology cubical complexes

were considered as an alternative to triangulations a new way to study top o

logical invariants combinatorially Later it turned out however that the cubical

cohomomology itself is not very advantageous in comparison with the simplicial

one Nevertheless as we see b elow cubical complexes as particular combinatorial

structures are very helpful in dierent geometrical and top ological considerations

Denitions and cubical maps

A q dimensional topological cube as a q ball with a face structure dened by

a homeomorphism with the standard q cub e I A face of a top ological q cub e is

thus the homeomorphic image of a face of I

Definition A nite top ological cubical complex is a subset C R

represented as a nite union U of top ological cub es of any dimensions called faces

in such a way that the following two conditions are satised

a Each face of a cub e in U b elongs to U

b The intersection of any two cub es in U is a face of each

The dimension of C is the maximal dimension of its faces The f vector of a cubical

complex C is f C f f where f is the numb er of ifaces

Remark The ab ove denition of cubical complex is a weaker cubical version

of Denition of geometrical simplicial complex If we replace top ological cub es

in Denition by convex p olytop es combinatorially equivalent to I then we

get the denition of a combinatorialgeometrical cubical complex or cubical polyhe

dron One can also dene an abstract cubical complex as a p oset more precisely a

semilattice such that each interval t is isomorphic to the face lattice of a cub e

We would not discuss here relationships b etween top ological geometrical and ab

stract cubical complexes since all examples we need for our further constructions

constitute a rather restricted family

The theory of f vectors of cubical complexes is parallel to a certain extent to

that of simplicial complexes but is much less develop ed It includes the notions

of hvector CohenMacaulay and Gorenstein cubical complexes and there are

cubical analogues of the UBC LBC and g conjecture See and for more

details A brief review of this theory and references can b e found in x

Since the combinatorial theory of cubical complexes is still in the early stage of

its development it may b e helpful to lo ok at some p ossible applications It turns

out that some particular problems from the discrete geometry and combinatorics

of cubical complexes arise naturally in statistical physics namely in connection

Ising model Since this asp ect is not widely known to with the dimensional

combinatorialists we make a brief digression to the corresp onding problems

CUBICAL COMPLEXES

q q

The standard unit cub e I together with all its faces is a q dimensional

cubical complex which we will also denote I Unlike simplicial complexes which

are always realizable as sub complexes in a simplex not every cubical complex

app ears as a sub complex of some I One example of a cubical complex not em

b eddable as a sub complex in any I is shown on Figure Moreover this complex

is not emb eddable into the standard cubical lattice in R for any q The authors

are thankful to M I Shtogrin for presenting this example

C H

A A

H A

Figure Cubical complex not emb eddable into cubical lattice

Problem S P Novikov Characterize k dimensional cubical complexes

C in particular cubical manifolds which admit

a a cubical embedding into the standard cubical lattice in R

b a map to the standard cubical lattice in R whose restriction to every k

dimensional cube is an isomorphism with a certain k face of the lattice

In the case when C is homeomorphic to a sphere the ab ove problem was solved

in Problem is an extension of the following question also formulated

Problem S P Novikov Suppose we are given a dimensional cubical

mo d cycle in the standard cubical lattice in R Describe al l maps of cubical

subdivisions of dimensional surfaces onto such that no two dierent squares are

mapped to the same square of

As it was told to the authors by S P Novikov the ab ove question was raised

during his discussions with A M Polyakov on the dimensional Ising mo del

Cubical sub divisions of simple p olytop es and simplicial complexes

Here we intro duce some particular cubical complexes which will play a piv

otal role in our further constructions in particular in the theory of momentangle

complexes We make no claims for originality of constructions app earing in this

most of them are part of mathematical folklore At the end we give section

several references to the sources where some similar considerations can b e found

All cubical complexes discussed here admit a canonical cubical emb edding into

the standard cub e To conclude the discussion in the end of the previous section we

note that the problem of emb eddability into the cubical lattice is closely connected

with that of emb eddability into the standard cub e For instance it is shown in

CUBICAL SUBDIVISIONS OF POLYTOPES AND SIMPLICIAL COMPLEXES

that if a cubical sub division of a dimensional surface is emb eddable into the

q q

standard cubical lattice in R then it also admits a cubical emb edding into I

Any face of I can b e written as

C fy y I y for i y for i g

q i i

where are two p ossibly empty subsets of q We set C C

m m

Construction canonical simplicial sub division of I Let

b e the simplex on the set m ie the collection of all subsets of m Assign to

each subset fi i g m the vertex v C of I More explicitly

v where if i and otherwise Regarding each

m i i

as a vertex of the barycentric sub division of we can extend the corresp ondence

v to a piecewise linear emb edding of the barycentric sub division into the

the b oundary complex of standard cub e I Under this emb edding denoted i

vertices of are mapp ed to the vertices of I having only one zero co ordinate while

the barycenter of is mapp ed to I see Figure The image i

is the union of m facets of I meeting at the vertex For each pair of

nonempty subsets of m all simplices of of the form

m m

are mapp ed to the same face C I The map i I extends to

cone by taking the vertex of the cone to I We denote the

resulting map by conei Its image is the whole I Hence conei cone

c c

I is a PL homeomorphism linear on the simplices of cone This denes a

triangulation of I which coincides with the canonical triangulation of the pro duct

of m onedimensional simplices see Construction It is also known as the

standard triangulation along the main diagonal

In short it can b e said that the canonical triangulation of I arises from the

m m

identication of I with the cone over the barycentric sub division of

n n

Construction cubical sub division of a simple p olytop e Let P R

Cho ose a p oint in the relative F b e a simple p olytop e with m facets F

interior of every face of P including the vertices and the p olytop e itself We get

the set S of f f f p oints here f P f f f is the

n n

f vector of P For each vertex v P dene the subset S S consisting of

the p oints chosen inside the faces containing v Since P is simple the numb er of

k faces meeting at v is 6 k 6 n Hence jS j The set S will b e the

v v

n n

can b e describ ed as The faces of C vertex set of an ncub e which we denote C

v v

k l l n k

follows Let G and G G Then there are b e two faces of P such that v G

lk l lk n k

The corresp onding p oints from and G exactly faces of P b etween G

We denote this face C S form the vertex set of an l k face of C Every

G G

1 2

n i

for some G G containing v The intersection of any two face of C is C

G G

1 2

n n i n

C is a face of each Indeed let G P b e the smallest face containing cub es C

n n n n

b oth vertices v and v Then C C C is the face of b oth I and I

0 0

i n

v v

G P

n n

Thereby we have constructed a cubical sub division of P with f P cub es of

dimension n We denote this cubical complex by C P

n m

There is an emb edding of C P to I constructed as follows Every n k

n n

n nk

face of P is the intersection of k facets G F F We map

i i

the corresp onding p oint of S to the vertex I where if

m i

i fi i g and otherwise This denes a mapping from the vertex set

k i

n m m

S of C P to the vertex set of I Using the canonical triangulation of I from

n m

Construction we extend this mapping to a PL emb edding i P I For P

CUBICAL COMPLEXES

Q A

i i

c c

A R

vertex of

cone

i i

c c

Figure Cone over as the standard triangulation of cub e

n n

P we have F each vertex v F

i i

n 1

n m

i C y y I y for j fi i g

P m j n

n m n

ie i C C I in the notation of The emb edding i P

P P

fi i g

1 n

I for n m is shown in Figure

We summarize the facts from the ab ove construction in the following statement

Proposition A simple polytope P with m facets can be split into cubes

n n n

C one for each vertex v P The resulting cubical complex C P embeds

canonical ly into the boundary of I as described by

CUBICAL SUBDIVISIONS OF POLYTOPES AND SIMPLICIAL COMPLEXES

n m

P C I C D

D B

A F A F E

n m

Figure The emb edding i P I for n m

Lemma The number of k faces of the cubical complex C P is given by

n n

f C P f P

k ni

n n

n n n

f P f P f P k n

n n k

k k

Proof This follows from the fact that the k faces of C P are in onetoone

n i

of emb edded faces of P G corresp ondence with the pairs G

Construction Let K b e a simplicial complex on m Then K is nat

m m

urally a sub complex of and K is a sub complex of As it follows from

Construction there is a PL emb edding i j jK j I The image i jK j

c K c

is an n dimensional cubical sub complex of I which we denote cubK We

have

C I cubK

ie cubK is the union of faces C I over all pairs of nonempty

simplices of K

Construction Since coneK is a sub complex of cone Con

struction also provides a PL emb edding

conei j j coneK j I

coneK

The image of this emb edding is an ndimensional cubical sub complex of I which

we denote ccK It can b e easily seen that

ccK C C

K K

the latter identity holds since C C C

Remark If fig m is not a vertex of K then ccK is contained in the

facet fy g of I i

CUBICAL COMPLEXES

The following statement summarizes the results of two previous constructions

Proposition For any simplicial complex K on the set m there is a

PL embedding of the polyhedron jK j into I linear on the simplices of K The

image of this embedding is the cubical subcomplex Moreover there is a PL

embedding of the polyhedron j coneK j into I linear on the simplices of coneK

whose image is the cubical subcomplex

A cubical complex C is called a cubical subdivision of cubical complex C if each

cub e of C is a union of nitely many cub es of C

Proposition For every cubical subcomplex C there exists a cubical sub

division that is embeddable into some I as a subcomplex

Proof Sub dividing each cub e of C as describ ed in Construction we obtain

Then applying Construction to K we get a a simplicial complex say K

C C

cubical complex cubK that sub divides K and therefore C It is emb eddable

C C

into some I by Prop osition

u u u

u u

u u u

a K p oints b K

Figure The cubical complex cubK

u u u

u u u u

u u u

a K p oints b K

Figure The cubical complex ccK

Example The cubical complex cubK in the case when K is a disjoint

union of vertices is shown in Figure a Figure b shows that for the

case K the b oundary complex of a simplex The corresp onding cubical

complexes ccK are indicated in Figure a and b

CUBICAL SUBDIVISIONS OF POLYTOPES AND SIMPLICIAL COMPLEXES

Remark As a top ological space cubK is homeomorphic to jK j while ccK

is homeomorphic to j coneK j On the other hand there is the cubical complex

cubconeK also homeomorphic to j coneK j However as cubical complexes

ccK and cubconeK dier since coneK coneK

Let P b e a simple np olytop e and K the corresp onding simplicial n

sphere the b oundary of the p olar simplicial p olytop e P Then ccK coincides

with the cubical complex C P from Construction More precisely ccK

i C P Thus Construction is a particular case of Construction compare

Figures

Remark Dierent versions of Construction can b e found in and in

some earlier pap ers listed there on p In p a similar construction was

intro duced while studying certain toric spaces we will return to this in the next

chapters A version of the cubical sub complex cubK I app eared in in

connection with Problem

CUBICAL COMPLEXES

CHAPTER

manifolds Toric and quasitoric

Toric varieties

Toric varieties app eared in algebraic geometry in the b eginning of s in con

nection with compactication problems for algebraic torus actions The geometry of

toric varieties very quickly has b ecome one of the most fascinating topics in algebraic

geometry and found applications in many mathematical sciences which otherwise

seemed far from each other We have already mentioned the pro of for the only if

part of the g theorem for simplicial p olytop es given by Stanley Other remarkable

applications include counting lattice p oints and volumes of lattice p olytop es rela

tions with Newton p olytop es and singularities after Khovanskii and Kushnirenko

discriminants resultants and hyp ergeometric functions after Gelfand Kapranov

and Zelevinsky reexive p olytop es and mirror symmetry for CalabiYau toric hy

p ersurfaces after Batyrev Standard references in the toric geometry are Danilovs

survey and b o oks by Oda Fulton and Ewald A more recent sur

vey article by Cox covers new applications including those mentioned ab ove

We are not going to give another review of the toric geometry here Instead in this

section we stress up on some top ological and combinatorial asp ects of toric varieties

We also give Stanleys argument for the g theorem

Toric varieties and fans Let C C n fg denote the multiplicative

group of complex numb ers The pro duct C of n copies of C is known as the

torus in the theory of algebraic groups In top ology the torus T is the pro duct

of n circles We keep the top ological notations referring to C as the algebraic

n n

torus The torus T is a subgroup of the algebraic torus C in the standard

way

n n i i

1 n

T e C e

where is running through R

Definition A toric variety is a normal algebraic variety M containing

as a Zariski op en subset in such a way that the natural the algebraic torus C

action of C on itself extends to an action on M

Hence C acts on M with a dense orbit

One of the most b eautiful prop erties of toric varieties is that all of their subtlest

algebrogeometrical prop erties can b e translated into the language of combinatorics

and convex geometry The following denition intro duces necessary combinatorial

notions

n n

Definition Fans terminology Let R b e the Euclidean space and Z

n n

R the integral lattice Given a nite set of vectors l l R dene the

TORIC AND QUASITORIC MANIFOLDS

convex polyhedral cone spanned by l l by

fr l r l R r > g

s s i

Any convex p olyhedral cone is a convex p olyhedron in the sense of Denition

Hence the faces of a convex p olyhedral cone are dened A cone is rational if

its generator vectors l l can b e taken from Z and is strongly convex if it

contains no line through the origin All cones considered b elow are strongly convex

and rational A cone is simplicial resp ectively nonsingular if it is generated by

n n n

a part of a basis of R resp ectively Z A fan is a set of cones in R such

that each face of a cone in is also a cone in and the intersection of two cones

in is a face of each A fan in R is called complete if the union of all cones

from is R A fan is simplicial resp ectively nonsingular if all cones of

are simplicial resp ectively nonsingular Let b e a simplicial fan in R with

m onedimensional cones or rays Cho ose generator vectors l l for these

rays to b e integer and primitive ie with relatively prime integer co ordinates The

fan denes a simplicial complex K on the vertex set m which is called the

underlying complex of By denition fi i g m is a simplex of K if

and only if l l span a cone of Obviously is complete if and only if

i i

K is a simplicial n sphere

As it is explained in any of the ab ove mentioned sources there is a onetoone

corresp ondence b etween fans in R and toric varieties of complex dimension n

We will denote the toric variety corresp onding to a fan by M It follows that

in principle all geometrical and top ological prop erties of a toric variety can b e

retrieved from the combinatorics of the underlying fan

The inclusion p oset of C orbits of M is isomorphic to the p oset of faces

of with reversed inclusion That is the k dimensional cones of corresp ond to

the co dimensionk orbits of the algebraic torus action on M In particular the

ndimensional cones corresp ond to the xed p oints while the origin corresp onds

to the unique dense orbit The toric variety M is compact if and only if is

complete If is simplicial then M is an orbifold ie is lo cally homeomorphic to

the quotient of R by a nite group action Finally M is nonsingular smo oth

if and only if is nonsingular which explains the notation Smo oth toric varieties

sometimes are called toric manifolds in the algebraic geometry literature

Remark Bistellar moves see Denition on the simplicial complex K

can b e interpreted as op erations on the fan On the level of toric varieties such

an op eration corresp onds to a ip a blowup followed by a subsequent blowdown

along dierent subvariety This issue is connected with the question of factoriza

tion of a prop er birational morphism b etween two complete smo oth or normal

algebraic varieties of dimension > into a sequence of blowups and blowdowns

with smo oth centers a fundamental problem in the birational algebraic geometry

Two versions of this problem are usually distinguished the Strong factorization

conjecture which asks if it is p ossible to represent a birational morphism by a

sequence of blowups followed by a sequence of blowdowns and the Weak factor

ization conjecture in which the order of blowups and blowdowns is insignicant

Since all toric varieties are rational any two toric varieties of the same dimension

are birationally equivalent Weak equivariant factorization conjecture for smo oth

lo darczyk announced in using complete toric varieties was proved by W

interpretation of equivariant ips on toric varieties as bistellar movetyp e op erations

TORIC VARIETIES

on the corresp onding fans Thereby the weak factorization theorem for smo oth toric

varieties reduces to the statement that any two complete nonsingular fans in R

can b e taken one to another by a nite sequence of bistellar movetyp e op erations

in which all intermediate fans are nonsingular This result is the essence of

Note that the statement do es not reduce to Pachners Theorem b ecause of

the additional smo othness condition The equivariant toric strong factorization

conjecture was proved by Morelli

Cohomology of nonsingular toric varieties The DanilovJurki

ewicz theorem allows us to read the integer cohomology ring of a nonsingular toric

variety directly from the underlying fan Write the primitive integer vectors along

the rays of in the standard basis of Z

l l l j m

j j nj

Assign to each vector l the indeterminate v of degree and dene linear forms

j j

l v l v Zv v 6 i 6 n

i i im m m

Denote by J the ideal in Zv v spanned by these linear forms ie J

The images of and J in the StanleyReisner ring ZK

n n

Zv v I see Denition will b e denoted by the same symb ols

m K

Theorem Danilov and Jurkiewicz Let be a complete nonsingular fan

in R and M the corresponding toric variety Then

a The Betti numbers the ranks of homology groups of M vanish in odd

dimensions while in even dimensions are given by

b M h K i n

i i

where hK h h is the hvector of K

b The cohomology ring of M is given by

H M Z Zv v I J ZK J

m K

where v 6 i 6 m denote the dimensional cohomology classes dual to invari

ant divisors codimensiontwo submanifolds D corresponding to the rays of

Moreover is a regular sequence in ZK

This theorem was proved by Jurkiewicz for pro jective smo oth toric varieties

and by Danilov Theorem in the general case Note that the ideal I is

determined only by the combinatorics of the fan ie by the intersection p oset of

while J dep ends on the geometry of One can observe that the rst part of

Theorem follows from the second part and Lemma

Remark As it was shown by Danilov the Q co ecient version of Theorem

is also true for simplicial fans and toric varieties

It follows from Theorem that the cohomology of M is generated by two

dimensional classes This is the rst thing to check if one wishes to determine

whether or not a given algebraic variety or smo oth manifold arises as a nonsingular

or simplicial toric variety Another interesting algebraicgeometrical prop erty of

nonsingular toric varieties suggested by Theorem is that the Chow ring

x of M coincides with its integer cohomology ring

TORIC AND QUASITORIC MANIFOLDS

Toric varieties from p olytop es

Construction Normal fan and toric varieties from p olytop es Supp ose

n n

we are given an np olytop e with vertices in the integer lattice Z R Such

a p olytop e is called integral or lattice Then the vectors l in 6 i 6 m can

b e chosen integer and primitive and the numb ers a can b e chosen integer Note

that l is normal to the facet F P and is p ointing inside the p olytop e P Dene

i i

the complete fan P whose cones are generated by those sets of normal vectors

l l whose corresp onding facets F F have nonempty intersection

i i i i

1 1

k k

in P The fan P is called the normal fan of P Alternatively if P then the

normal fan consists of cones over the faces of the p olar p olytop e P Dene the

toric variety M M The variety M is smo oth if and only if P is simple

P P

and the normal vectors l l of any set of n facets F F meeting at

i i i i

1 n 1 n

the same vertex form a basis of Z

Remark Every combinatorial simple p olytop e is rational that is admits a

convex realization with rational vertex co ordinates Indeed there is a small p er

turbation of dening inequalities in that makes all of them rational but do es

not change the combinatorial typ e since the halfspaces dened by the inequal

ities are in general p osition As a result one gets a simple p olytop e P of the

same combinatorial typ e with rational vertex co ordinates To obtain a realization

with integral vertex co ordinates we just take the magnied p olytop e k P for ap

propriate k Z We note that this is not the case in general in every dimension

> there exist nonrational convex p olytop es nonsimple and nonsimplicial see

eg Example and discussion there In dimension all convex p oly

top es are rational and in dimension the existence of nonrational p olytop es is an

op en problem Returning to simple p olytop es we note that dierent realizations of

a given combinatorial simple p olytop e as lattice p olytop es may pro duce dierent

even top ologically toric varieties M At the same time there exist combinatorial

simple p olytop es that do not admit any lattice realization with smo oth M We

present one such example in the next section see Example

The underlying top ological space of a toric variety M can b e identied with

n n

the quotient space T P for some equivalence relation using the following

construction see eg x

Construction Toric variety as an identication space We identify the

n n n n

torus T with the quotient R Z For each p oint q P dene Gq as the

smallest face that contains q in its relative interior The normal subspace to Gq

denoted N is spanned by the primitive vectors l see corresp onding to those

facets F which contain Gq If P is simple then there are exactly co dim Gq

such facets in general there are more of them Since N is a rational subspace

it pro jects to a subtorus of T which we denote T q Note that dim T q

n dim Gq Then as a top ological space

n n

M T P

where t p t q if and only if p q and t t T q The subtori T q

n n

are the isotropy subgroups for the action of T on M and P is identied with

n n

the orbit space Note that if q is a vertex of P then T q T so the vertices

n n

corresp ond to the T xed p oints of M At the other extreme if q int P

then T q feg so the T action is free over the interior of the p olytop e More

TORIC VARIETIES

generally if M P is the quotient pro jection then

int Gq T T q int Gq

Remark The ab ove construction can b e generalized to all complete toric va

rieties not necessarily coming from p olytop es by replacing P by an nball with

cellular decomp osition on the b oundary This cellular decomp osition is dual to

that dened by the complete fan

Construction allows us to dene the simplicial fan P and the toric va

riety M from any lattice simple p olytop e P However the lattice p olytop e P

contains more geometrical information than the fan P Indeed b esides the nor

mal vectors l which determine the fan we also have numb ers a Z 6 i 6 m

i i

see In the notations of Theorem it is well known in the toric geometry

that the linear combination D a D a D is an ample divisor on M

m m P

It denes a pro jective emb edding M C P for some r which can b e taken to b e

the numb er of vertices of P This implies that all toric varieties from p olytop es are

pro jective Conversely given a smo oth pro jective toric variety M C P one gets

very ample divisor line bundle D of a hyp erplane section whose zero cohomology is

generated by the sections corresp onding to lattice p oints in a certain lattice simple

a v a v H M Q p olytop e P For this P one has M M Let

m m P P

b e the cohomology class of D

Theorem Hard Lefschetz theorem for toric varieties Let P be a lat

tice simple polytope M the toric variety dened by P and a v

a v H M Q the above dened cohomology class Then the maps

m m P

ni ni

H M Q H M Q 6 i 6 n

P P

are isomorphisms

It follows from the pro jectivity that if M is smo oth then it is Kahler and

is the class of the Kahler form

Remark As it is stated Theorem applies only to simplicial pro jective

toric varieties However it remains true for any pro jective toric variety if we replace

the ordinary cohomology by the middle p erversity intersection cohomology For

more details see the discussion in x

Example The complex pro jective space C P fz z z z

n i

n n

C g is a toric variety The algebraic torus C acts on C P by

t t z z z z t z t z

n n n n

n n n

Obviously C C C P is a dense op en subset A sample fan dening

consists of the cones spanned by all prop er subsets of the set of n C P

vectors e e e e in R Theorem identies the cohomology

n n

ring H C P Z Zuu dim u with the quotient ring

Zv v v v v v v v

n n n n n

n n

The toric variety C P arises from a p olytop e C P M where P is the standard

nsimplex The corresp onding class H C P Q from Theorem is

represented by v

Now we are ready to give Stanleys argument for the only if part of the

g theorem for simple p olytop es

TORIC AND QUASITORIC MANIFOLDS

Proof of the necessity part of Theorem Realize the simple p oly

n n

top e as a lattice p olytop e P R Let M b e the corresp onding toric vari

ety Part a is already proved Theorem It follows from Theorem

that the multiplication by H M Q is a monomorphism H M Q

P P

H M Q for i 6 This together with part a of Theorem gives h 6 h

P i i

thus proving b To prove c dene the graded commutative Q 6 i 6

i i i

algebra A H M Q Then A Q A H M Q H M Q

P P P

for 6 i 6 and A is generated by degreetwo elements since so is H M Q

It follows from Theorem that the numb ers dim A h h 6 i 6

i i

are the comp onents of an M vector thus proving c and the whole theorem

Remark The DehnSommerville equations now can b e interpreted as the

Poincare duality for M Even though M needs not to b e smo oth the rational

P P

cohomology algebra of a simplicial toric variety or toric orbifold still satises the

Poincare duality

The Hard Lefschetz theorem Theorem holds only for pro jective toric vari

eties This implies that Stanleys argument cannot b e directly generalized b eyond

the p olytopal sphere case So far this case is the only generality in which meth

o ds involving the Hard Lefschetz theorem are ecient for proving the g theorem

see also the discussion at the end of section However the cohomology of

toric varieties has b een shown to b e quite helpful in generalizing statements like

the g theorem in a dierent direction namely to the case of general not neces

sarily simple or simplicial convex p olytop es So supp ose P is a convex lattice

np olytop e It gives rise as describ ed in Construction to a pro jective toric

variety M If P is not simple then M has worse than just orbifold singular

P P

ities and its ordinary cohomology b ehaves badly The Betti numb ers of M are

not determined by the combinatorial typ e of P and do not satisfy the Poincare

duality On the other hand it turns out that the dimensions h of the intersection

cohomology of M are combinatorial invariants of P The vector

b b b b

h P h h h

n n

is called the intersection hvector of P If P is simple then the intersection

hvector coincides with the ordinary one but in general h P is not determined

by the face vector of P and its combinatorial denition is quite subtle see

for details The intersection hvector satises the DehnSommerville equations

b b

h h and the Hard Lefschetz theorem shows that it also satises the GLBC

i ni

inequalities

b b b

h 6 h 6 6 h

n n

In the case when P cannot b e realized as a lattice p olytop e that is P is non

rational see the remark after Construction the combinatorial denition of

intersection hvector still works but it is not known whether the ab ove inequalities

continue to hold Some progress in this direction has b een achieved in

To summarise we may say that although Hard Lefschetz and intersection co

homology metho ds so far are not very helpful in the nonconvex situation like PL

or simplicial spheres they still are quite p owerful in the case of general convex

p olytop es

n n

Now we lo ok more closely at the action of the torus T C on a non

singular compact toric variety M This action is lo cally equivalent to the standard

QUASITORIC MANIFOLDS

n n

action of T on C see the next section for the precise denition The orbit space

M T is homeomorphic to an nball invested with the top ological structure of

manifold with corners by the xed p oint sets of appropriate subtori see x

Roughly sp eaking a manifold with corners is a space that is lo cally mo delled by

op en subsets of the p ositive cone R From this description it is easy to

deduce the strict denition which we omit here

n n

Construction Let P b e a simple p olytop e For any vertex v P

denote by U the op en subset of P obtained by deleting all faces not containing v

Obviously U is dieomorphic to R and even anely isomorphic to an op en set

n n

of R containing It follows that P is a manifold with corners with atlas fU g

As suggested by Construction if smo oth M arises from a lattice p oly

n n

top e P which is therefore simple then the orbit space M T is dieomorphic as

a manifold with corners to P Furthermore in this case there exists an explicit

n n n n

map M R the moment map with image P R and T orbits as bres

see x We will return to moment maps and some asp ects of symplectic

geometry in section The identication space description of a nonsingular

pro jective toric variety Construction motivated Davis and Januszkiewicz

to intro duce a top ological counterpart of the toric geometry namely the study of

quasitoric manifolds We pro ceed with their description in the next section

manifolds Quasitoric

Quasitoric manifolds can b e viewed as a top ological approximation to alge

braic nonsingular pro jective toric varieties This notion app eared in under

the name toric manifolds We use the term quasitoric manifold since toric

manifold is reserved in the algebraic geometry for nonsingular toric variety

In the consequent denitions we follow taking into account adjustments and

sp ecications from As in the case of toric varieties we rst give a denition of

a quasitoric manifold from the general top ological p oint of view as a manifold with

a certain nice torus action and then sp ecify a combinatorial construction similar

to the construction of toric varieties from fans or p olytop es

Quasitoric manifolds and characteristic maps As in the previous

n n

section we regard the torus T as the standard subgroup in C thereby

sp ecifying the orientation and the co ordinate subgroups T S i n

n n

in T We refer to the representation of T by diagonal matrices in U n as the

n n

standard action on C The orbit space of this action is the p ositive cone R

The canonical pro jection

n n n

T R C t t x x t x t x

n n n n

n n n

identies C with a quotient space T R This quotient will serve as the

lo cal mo del for some other identication spaces b elow

n n n

Let M b e a ndimensional manifold with an action of the torus T a T

manifold for short

Definition A standard chart on M is a triple U f where U is a

n n n

T stable op en subset of M is an automorphism of T and f is a equivariant

n n

homeomorphism f U W with some T stable op en subset W C The

n n

latter means that f t y tf y for all t T y U Say that a T action

TORIC AND QUASITORIC MANIFOLDS

n n n

on M is local ly standard if M has a standard atlas that is every p oint of M

lies in a standard chart

n n

The orbit space for a lo cally standard action of T on M is an ndimensional

manifold with corners Quasitoric manifolds corresp ond to the case when this orbit

space is dieomorphic as a manifold with corners to a simple p olytop e P Note

that two simple p olytop es are dieomorphic as manifolds with corners if and only

if they are combinatorially equivalent

n n

Definition Given a combinatorial simple p olytop e P a T manifold

n n

M is called a quasitoric manifold over P if the following two conditions are

satised

a the T action is lo cally standard

n n n

b there is a pro jection map M P constant on T orbits which maps

every k dimensional orbit to a p oint in the interior of a co dimensionk face

of P k n

n n

It follows that the T action on a quasitoric manifold M is free over the interior

n n n

of the quotient p olytop e P while the vertices of P corresp ond to the T xed

p oints of M Direct comparison with Construction suggests that every smo oth

pro jective toric variety M coming from a simple lattice p olytop e P is a qua

sitoric manifold over the corresp onding combinatorial p olytop e We will return to

this b elow in Example

Supp ose P has m facets F F By the denition for every facet F

m i

the preimage int F consists of co dimensionone orbits with the same

dimensional isotropy subgroup which we denote T F It can b e easily seen that

F is an n dimensional quasitoric submanifold over F with resp ect

i i

and refer to it as the facial to the action of T T F We denote it M

submanifold corresp onding to F Its isotropy subgroup T F can b e written as

i i

i n i

1i ni

T F e T e

t n

where R and Z is a primitive vector This is

i i ni i

determined by T F only up to a sign A choice of sign sp ecies an orientation

for T F For now we do not care ab out this sign and cho ose it arbitrarily A more

detailed treatment of signs and orientations is the sub ject of the next section We

refer to as the facet vector corresp onding to F The corresp ondence

i i

F T F

i i

is called the characteristic map of M

written as an intersection of k Supp ose we have a co dimensionk face G

facets G F F Then the submanifolds M M intersect

i i i i

1 1

k k

transversally in a submanifold M G which we refer to as the facial sub

manifold corresp onding to G The map T F T F T is injective

i i

since T F T F is identied with the k dimensional isotropy subgroup

i i

of M G It follows that the vectors form a part of an integral

i i

basis of Z

Let b e integer n mmatrix whose ith column is formed by the co ordinates

of the facet vector i m Every vertex v P is an intersection of n

facets v F F Let b e the maximal minor of formed

i i

v i i

1 n

QUASITORIC MANIFOLDS

by the columns i i Then

det

The corresp ondence

nk nk

G isotropy subgroup of M G

extends the characteristic map to a map from the face p oset of P to the

p oset of subtori of T

Definition Let P b e a combinatorial simple p olytop e and is a map

n n n

from facets of P to onedimensional subgroups of T Then P is called a

characteristic pair if F F T is injective whenever F F

i i i i

1 1

k k

The map directly extends to a map from the face p oset of P to the p oset of

n n n

subtori of T so we have subgroup G T for every face G of P

n n n n

As in the case of standard action of T on C there is a pro jection T P

n n

M whose bre over x M is the isotropy subgroup of x This argument can b e

used for reconstructing the quasitoric manifold from any given characteristic pair

Construction Quasitoric manifold from characteristic pair Given a

p oint q P we denoted by Gq the minimal face containing q in its relative

interior Now set

n n n

M T P

Gq compare with where t p t q if and only if p q and t t

n n n

Construction for toric varieties The free action of T on T P obviously

n n n

descends to an action on T P with quotient P The latter action is free

n n n

over the interior of P and has a xed p oint for each vertex of P Just as P is

covered by the op en sets U based on the vertices and dieomorphic to R see

n n n

Construction so the space T P is covered by op en sets T U

n n n n

and therefore to C This implies that the T homeomorphic to T R

n n n n

action on T P is lo cally standard and therefore T P is a quasitoric

manifold

n n

Definition Given an automorphism T T say that two qua

n n n

over the same P are equivariantly dieomorphic M sitoric manifolds M

n n

such that f t x tf x for M if there is a dieomorphism f M

n n

The automorphism induces an automorphism of the all t T x M

p oset of subtori of T Any such automorphism descends to a translation of

characteristic pairs in which the two characteristic maps dier by

The following prop osition is proved as Prop osition in see also

Prop osition

Proposition Construction denes a bijection between equivariant

dieomorphism classes of quasitoric manifolds and translations of pairs P

When is the identity we deduce that two quasitoric manifolds are equivari

antly dieomorphic if and only if their characteristic maps are the same

TORIC AND QUASITORIC MANIFOLDS

Cohomology of quasitoric manifolds The cohomology ring struc

ture of a quasitoric manifold is similar to that of a nonsingular toric variety To see

this analogy we rst describ e a cell decomp osition of M with even dimensional

cells a p erfect cellular structure and calculate the Betti numb ers accordingly

following

Construction We recall the Morsetheoretical arguments from the

pro of of DehnSommerville relations Theorem There we turned the

n n

skeleton of P into a directed graph and dened the index indv of a vertex v P

as the numb er of incident edges that p oint towards v These inward edges span a

face G of dimension ind v Denote by G the subset of G obtained by deleting all

v v v

ind v

faces not containing v Obviously G is dieomorphic to R and is contained in

G is identied with the op en set U P from Construction Then e

v v v

ind v n

C and the union of the e over all vertices of P dene a cellular decomp osi

tion of M Note that all cells are evendimensional and the closure of the cell e

indv n

is the facial submanifold M G M This argument was earlier used by

Khovanskii for constructing p erfect cellular decomp ositions of toric varieties

Proposition The Betti numbers of M vanish in odd dimensions while

in even dimensions are given by

n n

b M h P i n

i i

n n

where hP h h is the hvector of P

Proof The ith Betti numb er equals the numb er of idimensional cells in

the cellular decomp osition constructed ab ove This numb er equals the numb er of

vertices of index i which is h P by the argument from the pro of of Theorem

Given a quasitoric manifold M with characteristic map and facet vectors

t n

Z i m dene linear forms

i i ni

v v Zv v 6 i 6 n

i i im m m

The images of these linear forms in the StanleyReisner ring ZP will b e denoted

by the same letters

Lemma Davis and Januszkiewicz is a degreetwo regular

sequence in ZP

Let J denote the ideal in ZP generated by

Theorem Davis and Januszkiewicz The cohomology ring of M is

given by

n n

H M Z Zv v I J ZP J

m P

where v is the dimensional cohomology class dual to the facial submanifold

M with arbitrary orientation chosen i m

We give pro ofs for the ab ove two statements in section

Remark Change of sign of vector corresp onds to passing from v to v

i i i

in the description of the cohomology ring given by Theorem We will use this

observation in the next section

QUASITORIC MANIFOLDS

Nonsingular toric varieties and quasitoric manifolds In this

subsection we give a more detailed comparison of the two classes of manifolds

In general none of these classes b elongs to the other and the intersection of the

two classes contains smo oth pro jective toric varieties as a prop er sub class see

Figure Below in this subsection we provide the corresp onding examples

Quasitoric Smo oth pro jective Smo oth complete

manifolds toric varieties toric varieties

Figure

Example As it is suggested by comparing Constructions and

a nonsingular pro jective toric variety M arising from a lattice simple p olytop e

n n

P is a quasitoric manifold over the combinatorial typ e P The corresp onding

characteristic map F T F is dened by putting l in That is the

i i i i

facet vectors are the normal vectors l to facets of P i m see The

corresp onding characteristic n mmatrix is the matrix L from Construction

n n n

In particular if P is the standard simplex then M is C P Example

and E j where E is the unit n nmatrix and is the column of units

See also Example b elow

In general a smo oth nonpro jective toric variety may fail to b e a quasitoric

manifold although the orbit space for the T action is a manifold with corners

see section it may not b e dieomorphic or combinatorially equivalent to

a simple p olytop e The authors are thankful to N Strickland for drawing our

attention to this fact However we do not know of any such example

Problem Give an example of a nonsingular toric variety which is not

a quasitoric manifold

In p one can nd an example of a complete nonsingular fan in R

which cannot b e obtained by taking the cones with vertex over the faces of a con

vex simplicial p olytop e Nevertheless since the corresp onding simplicial complex

K is a simplicial sphere it is combinatorially equivalent to a p olytopal sphere

This means that the corresp onding nonsingular toric variety M although b e

ing nonpro jective is still a quasitoric manifold It is convenient to intro duce the

following notation here

Definition We say that a simplicial fan in R is strongly polytopal

or simply polytopal if it can b e obtained by taking the cones with vertex over

the faces of a convex simplicial p olytop e Equivalently a fan is strongly p olytopal

if it is a normal fan of simple lattice p olytop e see Construction Say that a

simplicial fan is weakly polytopal if the underlying simplicial complex K is a

p olytopal sphere that is combinatorially equivalent to the b oundary complex of a

simplicial p olytop e

TORIC AND QUASITORIC MANIFOLDS

Supp ose is a nonsingular fan and M the corresp onding toric variety Then

M is pro jective if and only if is strongly p olytopal and M is a quasitoric

manifold if and only if is weakly p olytopal Thus the answer to Problem

can b e given by providing a nonsingular fan which is not weakly p olytopal As

it was told to the authors by Y Civan in private communications this may

b e done by giving a singular fan whose underlying simplicial complex K is

the Barnette sphere see section and then desingularizing it using the standard

pro cedure see x Combinatorial prop erties of the Barnette sphere obstruct

the resulting nonsingular fan to b e weakly p olytopal

On the other hand it is easy to construct a quasitoric manifold which is not a

toric variety The simplest example is the manifold C P C P the connected sum

of two copies of C P It is a quasitoric manifold over the square I this follows

from the construction of equivariant connected sum see or section and

Corollary b elow However C P C P do not admit even an almost complex

structure ie its tangent bundle cannot b e made complex The following problem

arises

Problem Let P be a characteristic pair see Denition and

M the derived quasitoric manifold see Construction Find conditions

n n n

on P and so that M admits a T invariant complex or almost complex

structure

The almost complex case of the ab ove problem was formulated in Prob

lem Since every nonsingular toric variety is a complex manifold characteristic

pairs coming from lattice simple p olytop es as describ ed in Example provide

a sucient condition for Problem However this is not a necessary condition

even for the existence of an invariant complex structure Indeed there exist smo oth

nonpro jective toric varieties coming from weakly p olytopal fans see the already

mentioned example in p At the same time we do not know any example

of nontoric complex quasitoric manifold

Problem Find an example of a nontoric quasitoric manifold that ad

mits a T invariant complex structure

Although a general quasitoric manifold may fail to b e complex or almost com

plex it always admits a T invariant complex structure in the stable tangent bun

dle The corresp onding constructions are the sub ject of the next section We will

return to Problem in subsection

Another class of problems arises in connection with the classication of qua

sitoric manifolds over a given combinatorial simple p olytop e The general setting

of this problem is discussed in section Example b elow shows that there

are combinatorial simple p olytop es that do not admit a characteristic map and

therefore cannot arise as orbit spaces for quasitoric manifolds

Problem Give a combinatorial description of the class of polytopes P

that admit a characteristic map

A generalization of this problem is considered in chapter Problem

A characteristic map is determined by an integer n mmatrix which satis

es for every vertex v P The equation det denes a hyp ersur

face in the space M n m Z of integer n mmatrices

STABLY COMPLEX STRUCTURES AND COBORDISMS

Proposition The set of characteristic matrices coincides with the in

tersection

det

v P

of hypersurfaces in the space Mn m Z where v is running through the vertices

of the polytope P

Thus Problem is to determine for which p olytop es the intersection in

is nonempty

Example Example Let P b e a neighb orly simple p oly

n n

top e with m > facets eg the p olar of cyclic p olytop e C m with n > and

n n

m > see Example Then this P do es not admit a characteristic map

and therefore cannot app ear as the quotient space of a quasitoric manifold Indeed

by Prop osition it is sucient to show that intersection is empty Since

m > any matrix Mn m Z without zero columns contains two columns

say ith and j th which coincide mo dulo Since P is neighb orly the corre

sp onding facets F and F have nonempty intersection in P Hence the columns

i j

i and j of enter the minor for some vertex v P This implies that the

determinant of this minor is even and intersection is empty

In particular the ab ove example implies that there are no nonsingular toric

n n

varieties over the combinatorial p olar cyclic p olytop e C m with > facets

n n

This means that the combinatorial typ e C m with m > cannot b e realized

as a lattice simplicial p olytop e in such a way that the fan over its faces is non

singular In the toric geometry the question of whether for any given complete

simplicial fan there exists a combinatorially equivalent fan that gives rise

to a smo oth toric variety was known as Ewalds conjecture of The rst

counterexample was found in It was shown there that no fan over the faces

of a lattice realization of C m with m > n is nonsingular The comparison

of this result with Example suggests that some cyclic p olytop es C m with

small numb er of vertices b etween n and may app ear as the quotients of

quasitoric manifolds but not as quotients of nonsingular toric varieties

Another interesting corollary of Example is that the face ring ZC m

n n

of the b oundary complex of C m and the face ring ZpC m for any prime

p do es not admit a regular sequence of degree two or a lsop Of course since

ZpC m is CohenMacaulay it admits a nonlinear regular sequence Note

that in the case when k is of zero characteristic the ring kC m always admits

an lsop and degreetwo regular sequence by Lemma

Stably complex structures and quasitoric representatives in

cob ordism classes

This section is the review of results obtained by N Ray and the rst author

in and supplied with some additional comments

A stably complex structure on a smo oth manifold M is determined by a com

plex structure in the vector bundle M R for some k where M is the

tangent bund le of M and R denotes a trivial real k dimensional bundle over M

A stably complex manifold in other notations weakly almost complex manifold or

U manifold is a manifold with xed stably complex structure which one can view

where is a complex bundle isomorphic as a real bundle to as a pair M

TORIC AND QUASITORIC MANIFOLDS

M R for some k If M itself is a complex manifold then it p ossesses the

canonical stably complex structure M M The op erations of disjoint union

and pro duct endow the set of cob ordism classes M of stably complex manifolds

with the structure of a graded ring called the complex cobordism ring By the

theorem of Milnor and Novikov the complex cob ordism ring is isomorphic to the

p olynomial ring on an innite numb er of evendimensional generators

Za a deg a i

see The ring is the co ecient ring for generalized cohomology

theory known as the complex cobordisms We refer to as the standard source

for the cob ordism theory

Stably complex manifolds was the main sub ject of F Hirzebruchs talk at the

International Congress of Mathematicians see Using Milnor hypersur

faces Example and the MilnorNovikov theorem it was shown by Milnor

that every complex cob ordism class contains a nonsingular algebraic variety not

necessarily connected The following problem is still op en

Problem Hirzebruch Which complex cobordism classes in contain

connected nonsingular algebraic varieties

A weaker version of this question which is also op en asks which cob ordism

classes contain connected almost complex manifolds

Z is generated by Example The dimensional cob ordism group

the class of C P Riemannian sphere Every cob ordism class k C P

contains a nonsingular algebraic variety namely the disjoint union of k copies of

C P for k and a Riemannian surface of genus k for k 6 However

connected algebraic varieties are contained only in the cob ordism classes k C P

with k 6

The problem of cho osing appropriate generators for the ring is very imp or

tant in the cob ordism theory and its applications As it was recently shown in

and every complex cob ordism class of dimension contains a quasitoric

manifold see Theorem b elow By the denition quasitoric manifolds are nec

essarily connected so the result may b e considered as an answer to the quasitoric

analogue of Hirzebruchs question The construction of quasitoric representatives

in complex cob ordism classes relies up on an additional structure on a quasitoric

manifold called omniorientation which provides a combinatorial description for

canonical stably complex structures

n n

Let M P b e a quasitoric manifold with characteristic map Since

n n

the torus T is oriented a choice of orientation for P is equivalent to a choice

n n

An orientation of P is sp ecied by orienting the ambient of orientation for M

space R

Definition An omniorientation of a quasitoric manifold M consists

of a choice of an orientation for M and for every facial submanifold M

F i m

m n

Thus there are omniorientations in all for given M

An omniorientation of M determines an orientation for every normal bundle

M M i m Since every is a real plane bundle an

i i i

orientation of allows one to interpret it as a complex line bundle The isotropy i

STABLY COMPLEX STRUCTURES AND COBORDISMS

subgroup T F see of submanifold M F acts on the normal

i i

bundle i m Thus we have the following statement

Proposition A choice of omniorientation for M is equivalent to a

choice of orientation for P together with an unambiguous choice of facet vectors

i m in

We refer to a characteristic map as directed if all circles F i m

are oriented This implies that the signs of facet vectors

i i ni

i m are determined unambiguously In the previous section we organized

the facet vectors into the integer n m matrix This matrix satises

Due to the matrix carries exactly the same information as a directed

characteristic map Let Z denote the mdimensional free Zmo dule spanned by the

n F n

set F of facets of P Then denes an epimorphism Z Z by F

i i

F m

and an epimorphism T T which we will denote by the same letter In the

m F m F

sequel we write Z for Z and T for T assuming that the memb er e of the

m F

standard basis of Z corresp onds to the facet F Z i m and similarly

for T

Definition A directed characteristic pair P consists of a combina

torial simple p olytop e P and an integer matrix or equivalently an epimorphism

m n

Z Z that satises

Prop osition shows that the characteristic pair of an omnioriented quasitoric

manifold is directed On the other hand the quasitoric manifold derived from a

directed characteristic pair using Construction is omnioriented

Construction The orientation of the normal bundle over M de

i i

represented by nes an integral Thom class in the cohomology group H T

a complex line bundle over the Thom complex T We pull this back along the

The PontryaginThom collapse M T and denote the resulting bundle

i i

restriction of to M M is In the algebraic geometry this construction

i i i

corresp onds to assigning the line bundle to a divisor In particular in the case

when M is a smo oth toric variety the line bundle corresp onds to the divisor

D see Theorem

Theorem and Theorem Every omniorientation of a qu

asitoric manifold M determines a stably complex structure on it by means of the

fol lowing isomorphism of real mbund les

n mn

M R

The ab ove isomorphism of real vector bundles is essentially due to Davis and

Januszkiewicz see Theorem The interpretation of stably complex struc

tures in terms of omniorientations was given in

Corollary In the notation of Theorem suppose v H M is

the cohomology class dual to the oriented facial submanifold M of an omnioriented

quasitoric manifold M i m Then the total Chern class of stably

complex structure on M dened by the omniorientation is given by

n n

c M v v H M

It follows from Theorem that a directed characteristic pair P deter

n U

mines a complex cob ordism class M The following direct

TORIC AND QUASITORIC MANIFOLDS

extension of Theorem provides a description of the complex cob ordism ring of

an omnioriented quasitoric manifold

Proposition Prop osition Let v denote the rst cobordism

Chern class c M of the bund le 6 i 6 m Then the complex

i i

cobordism ring of M is given by

M v v I J

U m P

where the ideals I and J are dened in the same way as in Theorem

Note that the Chern class c is Poincare dual to the inclusion M

M by the construction of This highlights the remarkable fact that the complex

U n

b ordism groups M are spanned by embedded submanifolds By denition

n n n

the fundamental cob ordism class hM i M is dual to the b ordism class of

a p oint Thus hM i v v for any set fi i g such that F F

i i n i i

1 n 1 n

is a vertex of P

The following two examples are used to construct quasitoric representatives in

complex cob ordisms

Example b ounded ag manifold A bounded ag in C is a com

plete ag U fU U U C g for which U 6 k 6 n contains

n k

the co ordinate subspace C spanned by the rst k standard basis vectors As

it is shown in Example the ndimensional manifold B of all b ounded

n n

ags in C is a quasitoric manifold over the combinatorial cub e I with resp ect

n n

to the action induced by t z t z t z z on C where t T

n n n

Example A family of manifolds B 6 i 6 j is intro duced in

The manifold B consists of pairs U W where U is a b ounded ag in C see

j i j

C So B is a smo oth C P bundle Example and W is a line in U

over B It is shown in Example that B is a quasitoric manifold over the

i ij

i j

pro duct I

The canonical stably complex structures and omniorientations on the manifolds

B and B are describ ed in examples

n ij

n n

2 1

over p oly and M Remark The pro duct of two quasitoric manifolds M

n n n n

2 1 2 1

This construction P is a quasitoric manifold over P and P top es P

extends to omnioriented quasitoric manifolds and is compatible with stably com

plex structures details can b e found in Prop osition

It is shown in that the cob ordism classes of B multiplicatively gener

ate the ring Hence every ndimensional complex cob ordism class may b e

represented by a disjoint union of pro ducts

B B B

i j i j i j

1 1 2 2

k k

where i j k n Each such comp onent is a quasitoric manifold

q q

under the pro duct quasitoric structure This result is the substance of The

stably complex structures of pro ducts are induced by omniorientations and

are therefore also preserved by the torus action

To give genuinely quasitoric representatives which are by denition con

nected for each cob ordism class of dimension it remains only to replace the

disjoint union of pro ducts with their connected sum This is done in x

using Construction and its extension to omnioriented quasitoric manifolds

STABLY COMPLEX STRUCTURES AND COBORDISMS

Theorem Theorem In dimensions every complex cobor

dism class contains a quasitoric manifold necessarily connected whose stably com

plex structure is induced by an omniorientation and is therefore compatible with

the action of the torus

The connected sum op eration usually destroys the algebraicity of manifolds so

the complex cob ordism representatives provided by the ab ove theorem in general

are not algebraic compare Example

We note that in their work x Conner and Floyd constructed a class of

manifolds with canonical circle actions which can b e chosen as representatives for

multiplicative generators in oriented cob ordisms One can show that these Conner

Floyd manifolds can b e obtained as particular cases of manifolds B from Exam

ple However Conner and Floyd did not consider actions of halfdimensional

tori on their manifolds

Example The standard set of multiplicative generators for consists

i i j

of pro jective spaces C P i > and Milnor hypersurfaces H C P C P

6 i 6 j The hyp ersurface H is dened by

o n

i j

z z w w C P C P z w H

i j q q ij

However the hyp ersurfaces H are not quasitoric manifolds for i see

This can b e shown in the following way

i j

Construction Let C C b e the subspace spanned by the rst

j i

i vectors of the standard basis of C Identify C P with the set of complex

i j

lines l C To each line l assign the set of hyp erplanes C that contain l

j i

The latter set is identied with C P so there is a bundle E C P with bre

C P Here E is the set of pairs l l and the pro jection takes l to l

Lemma H is identied with the total space of bund le E C P

Proof A line l C is given by a vector z z z A hyp erplane

C is given by a linear form If we denote the co ecients of this linear form

by w w w then the condition l is exactly that from the denition of

Theorem The cohomology of H is given by

i j i k ik

H H Zu v u v u v

where deg u deg v

Proof We will use the notation from Construction Let denote the

i i j

bundle over C P whose bre over l C P is the j dimensional subspace l C

Then one can identify H with the pro jectivization C P Indeed for any line

l l representing a p oint in the bre of C P over l C P the hyp erplane

l C contains l so the pair l represents a p oint in H see

Lemma The rest of the pro of repro duces the general argument from the

Dold theorem ab out the cohomology of pro jectivizations

i i

Denote by the tautological line bundle over C P its bre over l C P is the

line l itself Then is a trivial j dimensional bundle Set w c

TORIC AND QUASITORIC MANIFOLDS

H C P Let c c c denote the total Chern class Since

c c and c w we get

c w w

Consider the pro jection p C P C P Denote by the tautological

line bundle over C P whose b er over a p oint l C P is the line l itself

Denote by the j bundle over C P whose bre over a p oint l l is the

orthogonal complement to l in l Then it is easy to see that p Set

v c H C P and u p w H C P Then u We have

c v and cp c c hence

c p c v u u v v

see But is j dimensional hence c Calculating the

homogeneous part of degree j in the ab ove identity we get the second identity

j i k ik

v u v

i j

Since b oth C P and C P have only evendimensional cells the LeraySerre

sp ectral sequence of the bundle p C P C P collapses at the E term It

follows that there is an epimorphism Zu v H C P and additively the

i j

cohomology of H C P coincides with that of C P C P Hence there are

no other relations except those two mentioned in the theorem

Proposition H is not a quasitoric manifold for i

Proof By Theorem the cohomology of a quasitoric manifold is isomor

phic to a quotient Zv v I J where the ideal I is generated by squarefree

monomials and J is generated by linear forms Due to we may assume without

loss of generality that rst n variables v v are expressed via the last m n

by means of linear equations with integer co ecients Hence we have

Zv v I J Zw w I

m mn

where I is an ideal having a basis each of whose elements is a pro duct of >

integer linear forms Supp ose now that H i is a quasitoric manifold Then

we have an isomorphism

Zw w I Zu v I

where I is the ideal from Theorem It is easy to see that in this case we have

m n ab ove and w w can b e identied with u v Thus the ideal I must

have a basis consisting of pro ducts of > linear forms with integer co ecients But

this is imp ossible for i

Combinatorial formulae for Hirzebruch genera of quasitoric

manifolds

The constructions from the previous section op en the way to evaluation of

cob ordism invariants Chern numb ers Hirzebruch genera etc on omnioriented

quasitoric manifolds in terms of the combinatorics of the quotient In this section

we exp ose the results obtained in this direction by the second author in

Namely using arguments similar to that from the pro of of Theorem we con

struct a circle action with only isolated xed p oints on any quasitoric manifold M

If M is omnioriented then this action preserves the stably complex structure and

HIRZEBRUCH GENERA OF QUASITORIC MANIFOLDS

its lo cal representations near xed p oints are describ ed in terms of the charac

teristic matrix This allows us to calculate Hirzebruchs genus as a sum of

contributions corresp onding to the vertices of p olytop e Each of these contributions

dep ends only on the lo cal combinatorics near the vertex In particular we obtain

formulae for the signature and the To dd genus of M

Definition The Hirzebruch genus asso ciated with the series

Qx q x q Q

k k

U n U

is the ring homomorphism Q that to each cob ordism class M

assigns the value given by the formula

n n

Qx hM i M

i Q

n n

Here M is a smo oth manifold whose stable tangent bundle M is a complex

bundle with complete Chern class in cohomology

c c c x

n i

and hM i is the fundamental class in homology

As it was shown by Hirzebruch every ring homomorphism Q arises

as a genus for some Q There is also an oriented version of Hirzebruch genera

O S

Q from the oriented cobordism which deals with ring homomorphisms

ring

Definition The genus is the Hirzebruch genus asso ciated with the

series

x y e

where y R is a parameter For particular values y we obtain the nth

Chern numb er the Todd genus and the Lgenus of M corresp ondingly

k k

Given a k dimensional oriented manifold X the signature signX is de

ned as the signature the numb er of p ositive squares minus the numb er of negative

ones of the intersection form

k k k

f hX i H X

k k

in the middledimensional cohomology H X We also extend the signature

to all evendimensional manifolds by setting signX It can b e shown

that the signature is multiplicative and is an invariant of cob ordism so it denes

Z a genus By the classical theorem of Hirze a ring homomorphism

bruch the signature coincides with the Lgenus and we will not distinguish

b etween the two notions in the sequel

n n

In the case when M is a complex manifold the value M can b e calcu

lated in terms of Euler characteristics of the Dolbeault complexes on M see

This was the original Hirzebruchs motivation for studying the genus

In this section we assume that we are given an omnioriented quasitoric manifold

n n

M over some P with characteristic matrix This sp ecies a stably complex

n n

structure on M as describ ed in the previous section The orientation of M

n n

determines the fundamental class hM i H M Z

TORIC AND QUASITORIC MANIFOLDS

The sign and index of a vertex edge vectors and calculation

of the genus Here we intro duce some combinatorial invariants of the torus

action and calculate the genus

Construction Supp ose v is a vertex of P expressed as the intersection

of n facets

v F F

i i

1 n

To each facet F ab ove assign the unique edge E such that E F v that

i k k i

k k

is E F Let e b e a vector along E with origin v Then e e is

k i k k n

j k

a basis of R which may b e either p ositively or negatively oriented dep ending on

the ordering of facets in Throughout this section we assume this ordering

to b e such that e e is a p ositively oriented basis

Once we sp ecied an ordering of facets in the facet vectors

i i

1 n

at v may in turn constitute either p ositively or negatively oriented basis dep ending

on the sign of the determinant of see

i i

1 n

Definition The sign of a vertex v F F is

i i

1 n

v det

One can understand the sign of a vertex geometrically as follows For each

n n

vertex v P the omniorientation of M determines two orientations of the

n n

tangent space T M at v The rst is induced by the orientation of M On the

other hand T M decomp oses into the sum of n twodimensional vector spaces

normal to the facial submanifolds M M containing v By the denition of

i i

1 n

omniorientation each of these twodimensional vector spaces is oriented so they

together dene another orientation of T M Then v if the two orientations

coincide and v otherwise

The collection of signs of vertices of P is an imp ortant invariant of an omnior

iented quasitoric manifold Note that reversing the orientation of M changes all

signs v to the opp osite At the same time changing the direction of one facet

vector reverses the signs for those vertices contained in the corresp onding facet

Let E b e an edge of P The isotropy subgroup of dimensional submanifold

E M is an n dimensional subtorus which we denote by T E It

can b e written as

n i i

1 n

T T E e e

n n

for some integers We refer to as the edge vector

n n

corresp onding to E This is a primitive vector in the dual lattice Z and is

determined by E only up to a sign There is no canonical way to cho ose these

signs simultaneously for all edges However the following lemma shows that the

omniorientation of M provides a canonical way to cho ose signs of edge vectors

lo cally at each vertex

Lemma For each vertex v P the signs of edge vectors

meeting at v can be chosen in such a way that the n nmatrix M

satises the identity

M E

where E is the unit matrix In other words and are conju

n i i

1 n

gate bases

HIRZEBRUCH GENERA OF QUASITORIC MANIFOLDS

Proof At the b eginning we cho ose signs of the edge vectors at v arbitrary

and express v as in Then is the edge vector corresp onding to the edge E

k k

opp osite to F k n It follows that E F for l k so T F T E

i k i i k

k l l

Hence

h i l k

k i

see and Since is a primitive vector and is a basis of

k i i

1 n

Z it follows from that h i Changing the sign of if necessary

k i i

k k

we obtain

h i

k i

which together with gives M E as needed

In the sequel while making some lo cal calculations near a vertex v we assume

that the signs of edge vectors are chosen as in the ab ove lemma It follows that the

edge vectors meeting at v constitute an integer basis of Z and

det M v

Example Supp ose M M is a smo oth toric variety arising from

a lattice simple p olytop e P dened by Then l i m see

i i

Example whereas the edge vectors at v P are the primitive integer vectors

e e along the edges with origin at v It follows from Construction that

v for any v compare with Prop osition b elow Lemma in this

case expresses the fact that e e and l l are conjugate bases of Z

n i i

1 n

Remark Globally Lemma provides two directions signs for an edge

vector one for each of its ends These two signs are always dierent if M is a

complex manifold eg a smo oth toric variety but in general they may b e the

same as well

t n

Let Z b e a primitive vector such that

h i for any edge vector

The vector denes a onedimensional oriented subtorus

n i i

1 n

T R T e e

Lemma Theorem For any satisfying the circle T

n n

acts on M with only isolated xed points corresponding to the vertices of P

For each vertex v F F the action of T induces a representation of S

i i

1 n

in the tangent space T M with weights h i h i

v n

Remark If M M is a smo oth toric variety then the genericity condi

tion is equivalent to that from the pro of of Theorem

Definition Supp ose we are given a primitive vector satisfying

Dene the index of a vertex v P as the numb er of negative weights of the

S representation in T M from Lemma That is if v F F then

v i i

1 n

ind v fk h i g

Remark The index of a vertex v can b e also dened in terms of the facet

vectors at v Indeed Lemma shows that if v F F then

i i

1 n

h i h i

i n i

1 n

TORIC AND QUASITORIC MANIFOLDS

Hence ind v equals the numb er of negative co ecients in the representation of

as a linear combination of basis vectors

i i

1 n

Theorem Theorem Theorem For any vector sat

isfying the genus of M can be calculated as

v n ind

v M y

v P

The pro of of this theorem uses the AtiyahHirzebruch formula and the circle

action from Lemma

Top Chern numb er and Euler characteristic The value of the

n n

genus M at y equals the nth Chern numb er c hM i for any

y y n

ndimensional stably complex manifold M If the stably complex structure

n n

on M comes from a complex structure in the tangent bundle ie if M is almost

complex then the nth Chern numb er equals the Euler characteristic of M

However for general stably complex manifolds the two numb ers may dier see

Example b elow

Given an omnioriented quasitoric manifold M Theorem gives the fol

lowing formula for its top Chern numb er

c M v

v P

n n

If M is a smo oth pro jective toric variety then v for every vertex v P

n n

see Example and c M equals the Euler characteristic eM Hence for

toric varieties the Euler characteristic equals the numb er of vertices of P which

of course is well known This is also true for arbitrary quasitoric M

n n

eM f P

To prove this one can just use Lemma and observe that the Euler characteristic

of an S manifold equals the sum of Euler characteristics of xed submanifolds

Comparing and we can deduce some results on the existence

n n

of a T invariant almost complex structure on a quasitoric manifold M see

Problem

An almost complex structure on M determines a canonical orientation of the

manifold A T invariant almost complex structure also determines orientations for

M i m since they are xed p oint the facial submanifolds M

sets for the appropriate subtori and thus gives rise to an omniorientation of M

Proposition Suppose that an omniorientation of a quasitoric manifold

n n

M is determined by a T invariant almost complex structure Then v for

any vertex v P and therefore

n n

c M eM

Proof Indeed the tangent space T M has canonical complex structure

and the orientations of normal subspaces to facial submanifolds meeting at v are

the canonical orientations of complex subspaces Hence the two orientations of

T M coincide and v

As a corollary we obtain the following necessary condition for the existence of

n n

a T invariant almost complex structure on M

HIRZEBRUCH GENERA OF QUASITORIC MANIFOLDS

n n

Corollary Let M be a quasitoric manifold over P and the corre

sponding characteristic matrix with undetermined signs of column vectors Sup

n n

pose M admits a T invariant almost complex structure Then the signs of col

umn vectors of can be chosen in such a way that the minors see are

positive for al l vertices v F F of P

i i

1 n

On the other hand due to a theorem of Thomas Theorem a real

orientable nbundle has a complex structure if and only if it has a stable complex

structure such that c e the latter denotes the Euler class It follows

from and that the condition from the ab ove corollary is also sucient

for a quasitoric manifold M to admit an almost complex structure not necessarily

T invariant Note that although the stably complex structure determined by

an omniorientation of a quasitoric manifold is T invariant see Theorem

the almost complex structure whose existence is claimed by the result of Thomas

n n

provided that the condition c M eM is satised may fail to b e invariant

Signature The value of the genus at y is the signature or the

Lgenus Theorem gives the following formula

Corollary The signature of an omnioriented quasitoric manifold M

can be calculated as

n ind v

signM v

v P

Being an invariant of an oriented cob ordism class the signature do es not de

p end on a particular choice of stably complex structure or omniorientation on

the oriented manifold M The following mo dication of Corollary provides

a formula for signM that do es not dep end on an omniorientation

Corollary Corollary The signature of an oriented quasi

toric manifold M can be calculated as

signM dete e

v P

where e k n are the edge vectors at v oriented in such a way that

he i

n n

If M M is a smo oth toric variety then v for any v P and

Corollary gives

ind v

signM

v P

Since in this case ind v equals the index from the pro of of Theorem we

obtain

signM h P

P k

Note that if n is o dd then the right hand side of the ab ove formula vanishes due

to the DehnSommerville equations The formula app ears in a more general

context in recent work of Leung and Reiner The quantity in the right hand

side of arises in the following combinatorial conjecture

TORIC AND QUASITORIC MANIFOLDS

Problem CharneyDavis conjecture Let K be a q dimensional

Gorenstein ag complex with hvector h h h Is it true that

h h h >

This conjecture was p osed in Conjecture D for ag simplicial homology spheres

Stanley Problem extended it to Gorenstein complexes The Charney

Davis conjecture is closely connected with the following dierentialgeometrical

conjecture

Problem Hopf conjecture Let M be a Riemannian manifold of non

positive sectional curvature Is it true that the Euler characteristic M satises

the inequality

q q

M >

More details ab out the connection b etween the CharneyDavis and Hopf con

jectures can b e found in and in a more recent pap er The relationships

b etween the ab ove two problems and the signature of a toric variety are discussed

To dd genus The next imp ortant particular case of genus is the

Todd genus corresp onding to y In this case the summands in the formula

from Theorem are not well dened for the vertices of index so it requires

some additional analysis

Theorem Theorem Theorem The Todd genus of

an omnioriented quasitoric manifold can be calculated as

td M v

v P ind v

the sum is taken over al l vertices of index

In the case of smo oth toric variety there is only one vertex of index This

is the b ottom vertex of P which has all incident edges p ointing out in the

notations used in the pro of of Theorem Since v for every v P

Theorem gives tdM which is well known see eg x Note

that for algebraic varieties the To dd genus equals the arithmetic genus

n n

If M is an almost complex manifold then td M > by Prop osition

and Theorem

Examples

Example Let us lo ok at the pro jective space C P regarded as a toric

variety Its stably complex structure is determined by the standard complex struc

Here ture in C P that is via the isomorphism of bundles C P C

C is the trivial complex line bundle and is the Hopf line bundle over C P The

orientation is dened by the complex structure The toric variety C P arises from

the dimensional lattice simplex with vertices and The facet

vectors here are primitive normal to facets and p ointing inside the p olytop e The

edge vectors are primitive parallel to edges and p ointing out of the corresp onding

vertex This is shown in Figure Let us calculate the To dd genus and the signa

ture using Corollary and Theorem We have v v v

HIRZEBRUCH GENERA OF QUASITORIC MANIFOLDS

Take then indv indv ind v rememb er that the

index is the numb er of negative scalar pro ducts of edge vectors with Thus

signC P signC P td C P td C P

= ( 1 1)

= (1 0)

v v

= (0 1)

Figure C P C

Example Now consider C P with the omniorientation determined by

the three facet vectors shown in Figure This omniorientation diers

from the previous example by the sign of The corresp onding stably complex

structure is determined by the isomorphism C P R Using

we calculate

v v v

Taking we nd ind v ind v ind v Thus

signC P tdC P

Note that in this case formula gives

c C P v v v

while the Euler numb er of C P is

T equivariant stably complex and almost complex manifolds were considered

in works of Hattori and Masuda as a separate generalization called the

unitary toric manifolds of toric varieties Instead of Davis and Januszkiewiczs

characteristic maps Masuda in used the notion of multifan to describ e the

combinatorial structure of the orbit space The multifan is a collection of cones

which may overlap unlike a usual fan The To dd genus of a unitary toric manifold

was calculated in via the degree of the overlap of cones in the multifan This

result is equivalent to our Theorem in the case of quasitoric manifolds A

formula for the genus similar to that from Theorem has b een obtained

indep endently in a more recent pap er For more information ab out multi

fans see

TORIC AND QUASITORIC MANIFOLDS

= (1 1)

= (1 0)

v v

= (0 1)

Figure C P C

Classication problems

There are two main classication problems for quasitoric manifolds over a given

simple p olytop e the equivariant ie up to a equivariant dieomorphism and

the top ological ie up to a dieomorphism Due to Prop osition the equi

variant classication reduces to describing all characteristic maps for given simple

p olytop e P The top ological classication problem usually requires additional

analysis In general b oth problems seem to b e intractable However in some par

ticular cases nice classication results may b e achieved Here we give a brief review

of what is known on the sub ject

n n

Let M b e a quasitoric manifold over P with characteristic map We

assume here that the facets are ordered in such a way that the rst n of them share

a common vertex

Lemma Up to equivalence see Denition we may assume that

F is the ith coordinate subtorus T T i n

i i

Proof Since the onedimensional subtori F i n generate T

we may dene as any automorphism of T that maps F to T

i i

It follows that M admits an omniorientation whose corresp onding characteristic

n mmatrix has the form E j where E is the unit matrix and denotes some

integer n m nmatrix

n n

In the simplest case P the equivariant and top ological classication

of quasitoric manifolds reduces to the following easy result

Proposition Any quasitoric manifold over is equivariantly dif

feomorphic to C P regarded as a toric variety see Examples and

Proof The characteristic map for C P has the form

n n

F T i n F S

C P i i C P n d

i i n n

where S fe e T g R is the diagonal subgroup in T Let

n n

M b e a quasitoric manifold over with characteristic map We may assume M

CLASSIFICATION PROBLEMS

that F T i n by Lemma Then it easily follows from

M i i

that

i i n

1 n

F e e T R

M n

n n

where i n Now dene the automorphism T T by

i i i i

1 n 1 1 n n

e e e e

It can b e readily seen that which together with Prop osition

M C P

completes the pro of

Both problems of equivariant and top ological classication also admit a com

plete solution for n ie for quasitoric manifolds over p olygons

Example Given an integer k the Hirzebruch surface H is the complex

manifold C P C where is the complex line bundle over C P with rst

k k

Chern class k and C P denotes the pro jectivisation of a complex bundle In

particular each Hirzebruch surface is the total space of the bundle H C P

with bre C P The surface H is dieomorphic to S S for even k and to

C P C P for o dd k where C P denotes the space C P with reversed orientation

Each Hirzebruch surface is a nonsingular pro jective toric variety see p The

orbit space for H regarded as a quasitoric manifold is a combinatorial square

the corresp onding characteristic maps can b e describ ed using Example see

also Example

Theorem p A quasitoric manifold of dimension is equiv

ariantly dieomorphic to an equivariant connected sum of several copies of C P and

Hirzebruch surfaces H

Corollary A quasitoric manifold of dimension is dieomorphic to a

connected sum of several copies of C P C P and S S

The classication problem for quasitoric manifolds over a given simple p olytop e

can b e considered as a generalization of the corresp onding problem for nonsingular

toric varieties The classication result for dimensional toric varieties is similar

to Theorem and can b e found eg in In to every toric variety

over a simple p olytop e P there were assigned two integer weights on every edge

of the dual simplicial complex K Using the sp ecial mono dromy conditions

for weights the complete classication of toric varieties over simple p olytop es

with 6 facets was obtained in A similar construction was used in to

obtain the classication of toric varieties over P with m n facets note that

any such simple p olytop e is a pro duct of two simplices

In the construction of weights from was generalized to the case of

quasitoric manifolds This resulted in a criterion Theorem for the existence

of a quasitoric manifold with prescrib ed weight set and signs of vertices see Def

inition The metho ds of allow one to simplify the equations for

characteristic map on a given p olytop e As an application results on the classi

cation of quasitoric manifolds over a pro duct of an arbitrary numb er of simplices

were obtained there

TORIC AND QUASITORIC MANIFOLDS

CHAPTER

tangle complexes Momen

Momentangle manifolds Z dened by simple p olytop es

For any combinatorial simple p olytop e P with m facets Davis and Janusz

m n

kiewicz intro duced in a T manifold Z with orbit space P This manifold

n n

has the following universal prop erty for every quasitoric manifold M P

mn n

there is a principal T bundle Z M whose comp osite map with is the

orbit map for Z Top ology of manifolds Z and their further generalizations is

P P

very nice itself and at the same time provides an eective to ol for understanding

interrelations b etween algebraic and combinatorial ob jects such as StanleyReisner

rings subspace arrangements cubical complexes etc In this section we repro duce

the original denition of Z and adjust it in a way convenient for subsequent

generalizations

Let F fF F g b e the set of facets of P For each facet F F denote

m i

F m

by T the onedimensional co ordinate subgroup of T T corresp onding to F

F i

Then assign to every face G the co ordinate subtorus

T T T

G F

F G

Note that dim T co dim G Recall that for every p oint q P we denoted by

Gq the unique face containing q in the relative interior

Definition For any combinatorial simple p olytop e P intro duce the iden

tication space

F n

Z T P

T where t p t q if and only if p q and t t

Remark The ab ove denition resembles constructions and but this

time the equivalence relation dep ends only on the combinatorics of P Similar

constructions app eared in earlier works of Vinb erg and Davis on reection

groups

m F n

The free action of T on T P descends to an action on Z with quo

n n m

tient P Let Z P b e the orbit map The action of T on Z is free

P P

n n

over the interior of P while each vertex v P represents the orbit v with

maximal isotropy subgroup of dimension n

Lemma The space Z is a smooth manifold of dimension m n

We will provide several dierent pro ofs of this lemma each of which arises from

an equivalent denition of Z To give our rst pro of we need the following simple

top ological fact

k k

Proposition The torus T admits an embedding into R

MOMENTANGLE COMPLEXES

Proof The statement is obvious for k Supp ose it holds for k i

i i i

We may assume that T is emb edded into an iball D R Represent the

i i i

i sphere as S D S S D two pieces are glued by the identity

i i

dieomorphism of the b oundaries By the assumption the torus T T S

i i i i

can b e emb edded into D S and therefore into S Since T is compact and S

i i i

is the onep oint compactication of R we have T R and the statement

follows by induction

Proof of Lemma Construction provides the atlas fU g for P as a

manifold with corners The set U is based on the vertex v and is dieomorphic

n mn n mn n

to R Then U T R We claim that T R can b e realized

as an op en set in R thus providing a chart for Z To see this we emb ed

mn mn

T into R as a closed hyp ersurface H Prop osition Since the nor

mal bundle is trivial a small neighb orho o d of H R is homeomorphic to

mn n

T R Taking the cartesian pro duct with R we obtain an op en set in

mn mn n

R homeomorphic to T R

The following statement follows easily from the denition of Z

Proposition If P P P for some simple polytopes P P then

Z Z Z If G P is a face then Z is a submanifold of Z

P P P G P

1 2

n n

Supp ose now that we are given a characteristic map on P and M is the

derived quasitoric manifold Construction Cho osing an omniorientation in

F n

any way we obtain a directed characteristic map T T Denote its kernel by

H it dep ends only on then H is an m ndimensional subtorus of T

Proposition The subtorus H acts freely on Z thereby dening a

mn n

principal T bund le Z M

Proof It follows from that H meets every isotropy subgroup only at

the unit This implies that the action of H on Z is free By denitions of Z

P P

n F n n n

and M the pro jection id T P T P descends to the pro jection

F n n n

T P T P

mn n

which displays Z as a principal T bundle over M

m m F

To simplify notations from now on we will write T C etc instead of T

C etc

Consider the unit p olydisc D in the complex space

m m

D z z C jz j 6 i m

m i

m m m

Then D is stable under the standard action of T on C and the quotient is

m m

the unit cub e I R

n m

Lemma The cubical embedding i P I from Construction is

Z D covered by an equivariant embedding i

P e

n n

Proof Recall that the cubical complex C P consists of the cub es C based

n n n

on the vertices v P Note that C is contained in the op en set U P see

Construction The inclusion C U is covered by an equivariant inclusion

m n n

B C where B C is a closed subset homeomorphic to D

v v

m mn

B and B is stable under the T action the resulting T Since Z

v v P

v P

emb edding Z D is equivariant P

GENERAL MOMENTANGLE COMPLEXES Z

It follows from the pro of that the manifold Z is represented as a union of

f P closed T invariant subspaces B In section we will use this to con

n v

struct a cell decomp osition of Z For now we mention that if v F F

P i i

1 n

then

n m

i B D T D

e v

i i

1 n i i g mnf

1 n

or more precisely

i B z z D jz j for i fi i g

e v m i n

Recalling that the vertices of P corresp ond to the maximal simplices in the p oly

topal sphere K the b oundary of p olar p olytop e P we can write

i Z D T D

e P

where b m n The ab ove formula may b e regarded as an alternative de

nition of Z Intro ducing the p olar co ordinates in D we see that i B is

P e v

parametrized by n radial or moment and m angle co ordinates We refer to Z as

the momentangle manifold corresponding to P

n n

Example Let P the nsimplex Then Z is homeomorphic

n n

to the n sphere S The cubical complex C see Construction

n n n

is homeomorphic to D Each subset B C consists of n cub es C

v v

S In particular for n we obtain the representation of the sphere S as a

union of two solid tori D S and S D glued by the identity dieomorphism

of their b oundaries

Another way to construct an equivariant emb edding of Z into C can b e

derived from Construction

n m

dened Construction Consider the ane emb edding A P R

by It is easy to see that Z enters the following pullback diagram

Z C

y y

n m

P R

Thus there is an equivariant emb edding Z C covering A A choice of

P P

matrix W in Construction gives a basis in the m ndimensional subspace

orthogonal to the nplane containing A P see The following statement

follows

Corollary see also x The embedding Z C has the trivial

normal bund le In particular Z is nul lcobordant

Remark Another way to see that Z is nullcob ordant is to establish a free

S action on it see eg Prop osition Then we get the manifold Z D

with b oundary Z

General momentangle complexes Z

m m

In this section for any cubical sub complex in I we dene a certain T stable

sub complex in the mdisc D In particular this provides an extension of the

construction of Z to the case of general simplicial complex K The resulting space P

MOMENTANGLE COMPLEXES

denoted Z is not a manifold for arbitrary K but is so when K is a simplicial

sphere The complex Z as a generalization of manifold Z rst app eared in

K P

x The approach used there involves the notion of simple p olyhedral complex

which extends the corresp ondence b etween p olytopal simplicial spheres and simple

p olytop es to general simplicial complexes

m m

In the sequel we denote the canonical pro jection D I and any of its

m m m

restriction to a closed T stable subset of D by For each face C of I

see dene

B C

fz z D z for i jz j for i g

m i i

j i mj

It follows that if j j i and j j j then B D T where the disc

j i mj

factors D D are indexed by n while the circle factors S T are

indexed by m n

Definition Let C b e a cubical sub complex of I The momentangle

complex maC corresp onding to C is the T invariant decomp osition of C

into the momentangle blo cks B corresp onding to the faces C of C

Thus maC is dened from the commutative diagram

maC D

y y

C I

The torus T acts on maC with orbit space C

Let K b e a simplicial complex on the set m In section two canonical

cubical sub complexes of I namely cubK and ccK were asso ciated

to K We denote the corresp onding momentangle complexes by W and Z

K K

resp ectively Thus we have

m m

Z D W D

K K

and

y y y y

m m

ccK I cubK I

where the horizontal arrows are emb eddings while the vertical ones are orbit maps

for T actions Note that dim Z m n and dim W m n

K K

Remark Supp ose that K K for some simple p olytop e P Then it follows

from that Z is identied with Z or more precisely with i Z The

K P e P

simple p olyhedral complex P used in to dene Z for general K now can

K K

b e interpreted as a certain face decomp osition of the cubical complex ccK see

also the pro of of Lemma b elow

Note the complex Z dep ends on the ambient set m of K as well as the

complex K In the case when it is imp ortant to emphasize this we will use the

notation Z If we assume that K is a simplicial complex on the vertex set m

then Z is determined by K However in some situations see eg section it is

convenient to consider simplicial complexes K on m whose vertex sets are prop er

subsets of m Let fig b e a ghost vertex of K ie fig is a oneelement subset of

m which is not a vertex of K Then the whole cubical sub complex ccK I is

CELL DECOMPOSITIONS OF MOMENTANGLE COMPLEXES

contained in the facet fy g of I see the remark after Construction The

following prop osition follows easily from

Proposition Suppose fi g fi g are ghost vertices of K Then

Z Z T

Km Kmnfi i g

We call this easy observation stabilization of momentangle complexes via

the multiplication by tori It means that if we emb ed K into a set larger than

its vertex set then the corresp onding complex Z is multiplied by the torus of

dimension equal to the numb er of ghost vertices

Example Let K b e the b oundary of m simplex Then ccK

is the union of m facets of I meeting at the vertex and Z is the

m sphere S compare with Example

Let K b e an m simplex Then ccK is the whole cub e I and Z is

the mdisc D

Lemma Suppose K is a simplicial n sphere Then Z is an m n

dimensional closed manifold

Proof In this pro of we identify the p olyhedrons jK j and j coneK j with

m m m

their images cub K I and ccK I under the map j coneK j I see

Prop osition For each vertex fig K denote by F the union of n cub es

e e e

of cubK that contain fig Alternatively F is j star figj These F F will

i K m

play the role of facets of a simple p olytop e If K K for some P then F is the

P i

image of a facet of P under the map i C P I see Construction As in

the case of simple p olytop es we dene faces of ccK as nonempty intersections

e e

of facets F F Then the vertices ie nonempty intersections of n

facets are the barycenters of n simplices of jK j For every such barycenter

b denote by U the op en subset of ccK obtained by deleting all faces not

while U is homeomorphic to containing b Then U is identied with R

b b

mn n

T R This denes a structure of manifold with corners on the nball ccK

j coneK j with atlas fU g Furthermore Z ccK is a manifold with

b K

atlas f U g

Problem Characterise simplicial complexes K for which Z is a man

ifold

We will see b elow Theorem that if Z is a manifold then K is a Goren

stein complex see Denition for homological reasons Hence the answer

to the ab ove problem is somewhere b etween simplicial spheres and Gorenstein

complexes

Cell decomp ositions of momentangle complexes

Here we consider two cell decomp ositions of D and apply them to construct

cell decomp ositions for momentangle complexes The rst one has cells and

descends to a cell complex structure with typ es of cells on any momentangle

m m m

complex maC D The second cell decomp osition of D has only

cells but it denes a cell complex structure with typ es of cells only on moment

angle complexes Z

Let us consider the cell decomp osition of D with one cell D two cells I T

and two cells shown on Figure a It denes a cell complex structure

MOMENTANGLE COMPLEXES

D D

T I T

s s s

a b

Figure Cell decomp ositions of D

m m

on the p olydisc D with cells Each cell of this complex is a pro duct of

cells of dierent typ es D I T and i m We enco de cells

i i i i i

in the language of sign vectors used in the theory of hyp erplane arrangements

m m

see eg Each cell of D with resp ect to our cell decomp osition will

b e represented by a sign vector R fD I T g We denote by R R R

D I

and R resp ectively the D I T and comp onent of R Each of these R

comp onents can b e seen as a subset of m and all ve subsets are complementary

This justies the notations jR j jR j jR j jR j and jR j for the numb er of D

D I T

I T and entries of R resp ectively In particular we see that the closure

of a cell R is homeomorphic to a pro duct of jR j discs jR j segments and jR j

D I T

circles Our rst observation is that this cell decomp osition of D induces a cell

decomp osition of any momentangle complex in D

Lemma For any cubical subcomplex C of I the corresponding moment

angle complex maC is a cel lular subcomplex of D

Proof Indeed maC is a union of momentangle blo cks B and

each B is the closure of cell R with R n R R m n

D T

R R

Now we restrict our attention to the momentangle complex Z corresp onding

to cubical complex ccK I see By the denition Z is the union of

momentangle blo cks B D with K Denote

B B z z D jz j for j

m j

Then B C rememb er our previous notation C C and B B

for any It follows that

Z B

compare this with the note after

Remark If K K for a simple p olytop e P and j j n then B is i B

P e v

F Hence reduces to in this case for v

0 0

This observation allows us to simplify the cell Note that B B B

decomp osition from Lemma in the case maC Z For this we replace the

union of cells I D see Figure a by one dimensional cell which we keep

denoting D for simplicity The resulting cell decomp osition of D with cells is

m m

shown on Figure b It denes a cell decomp osition of D with cells

CELL DECOMPOSITIONS OF MOMENTANGLE COMPLEXES

each of which is a pro duct of dierent typ es D T and i m Again

i i i

we use the sign vector language and enco de the new cells of D by sign vectors

T fD T g The notation T jT j etc have the same meaning as in the case

D T

of cell decomp osition The closure of T is now a pro duct of jT j discs and jT j

D T

circles

Lemma The momentangle complex Z is a cel lular subcomplex of D

with respect to the cel l decomposition see Figure b Those cel ls T

D which form Z are determined by the condition T K

K D

Proof Since B B is the closure of cell T with T T m n

D T

and T the statement follows from

Remark Note that for general C the momentangle complex maC is not a

m m

cell sub complex with resp ect to the cell decomp osition of D

Lemma Let K K be an inclusion of simplicial complexes on

the sets m and m respectively Then it induces an equivariant cel lular map

Z Z of the corresponding momentangle complexes

K K ma

1 2

Proof Assign the ith vector of the standard basis of C to the element

i m and similarly for C and m This allows us to extend the map

m m

1 2

m m to an inclusion C C For any subset m the map

m m

1 2

takes B C see to B C Since is a simplicial map for a

simplex of K we have K and B Z see Hence denes

C K C

an equivariant map Z Z

ma K K

1 2

Let us apply the construction of Z see to the case when K

regarded as a simplicial complex on m Then ccK I the cone

over the empty set is just one vertex and so Z T where

Z Z Another way to see this is to apply Prop osition in the case

K We also observe that Z is contained in Z for any K on m as a

T stable subset

Lemma Let K be a simplicial complex on the vertex set m The inclu

sion Z Z is a cel lular embedding homotopical to a map to a point ie the

torus Z is a cel lular subcomplex contractible within Z

Proof Z Z is a cellular sub complex since it is the closure of the m

dimensional cell T with T m So it remains to prove that T is contractible

within Z We do this by induction on m If m then the only option for

K is K simplex so Z D see Example and Z S is

contractible in Z Now supp ose that the vertex set of K is m The emb edding

under the question factors as

Z Z Z

m K m Km

[ m1]

where K is the maximal sub complex of K on the vertex set m see

By Prop osition Z Z S and Z Z

m m K m K m

[m1] [m1]

S By the inductive hyp othesis we may assume that the emb edding Z

Z is nullhomotopic so the comp osite emb edding is homotopic to

K m

[m1]

the map Z S Z that sends Z to a p oint and S to the

m Km m

closure of the cell T Z the latter is understo o d as a vector of

letters D T But since fmg is a vertex of K the complex Z also contains the K

MOMENTANGLE COMPLEXES

cell D so a disc D is patched to the closure of T It follows

that the whole map is nullhomotopic

Corollary For any simplicial complex K on the vertex set m the

momentangle complex Z is simply connected

Proof Indeed the skeleton of our cellular decomp osition of Z is contained

in the torus Z which is nullhomotopic by Lemma

Momentangle complexes corresp onding to joins connected sums

and bistellar moves

Here we study the b ehavior of momentangle complexes Z with resp ect to

constructions from section In particular we describ e momentangle complexes

corresp onding to joins and connected sums of simplicial complexes and interpret

bistellar moves see Denition as certain surgerylike op erations on moment

angle complexes

Agreement Let fj j g b e a subset of m In this section we will

mk k mk k

T The denote the momentangle blo ck B D T by D

b oundary of B is

mk k mk

T S T B D

nfj g

k mk

compare with Example We denote B S T Furthermore for

any partition m into three complementary subsets with j j i j j j

r j i

for the corresp onding subset T S jj r we will use the notation D

of D

Construction momentangle complex corresp onding to join Let K

K b e simplicial complexes on the sets m m resp ectively and K K the join

m m m m

1 2 1 2

of K and K see Construction Identify the cub e I with I I

Then using we calculate

ccK K C C C

1 2 1 2

K K K K

1 1 2 2 1 1 2 2

C C ccK ccK

1 2

K K

1 1 2 2

Hence

Z Z Z

K K K K

1 2 1 2

This can b e thought as a generalization of Prop osition to arbitrary simplicial

complexes

Construction momentangle complexes and connected sums Supp o

se we are given two pure n dimensional simplicial complexes K K on the

sets m m resp ectively and let K K b e their connected sum at some

and Here K K is considered as a simplicial complex on m m n

with suitable identication see Construction If we regard K

as a simplicial complex on m m n then the corresp onding momentangle

JOINS CONNECTED SUMS AND BISTELLAR MOVES

m n

complex is Z T Prop osition where Z Z and similarly

K K

K m

1 1

b b

for K Denote K K n f g and K K n f g Then

n m n m n n

2 1

Z n D T Z n T D Z Z

K K

b b

2 1

K K

2 1

by Now we see that

m n m n

2 1

T T Z Z Z

K K

b b

1 2

K K

2 1

m n n m n m n n

1 2 1

where the two pieces are glued along T S T T S

1 2

m n

T using the identication of with Equivalently

j j m m nj j

1 2

Z D T

K K

1 2

K or K

1 2

Example Let K K b e a pure n dimensional simplicial complex

on m and K the b oundary of nsimplex Cho ose a maximal simplex

K and consider the connected sum K the choice of a maximal simplex

n n

in is irrelevant Note that Z S can b e decomp osed as

n n

2n1 1

D S S D

S S

see examples and therefore Z S D Now it follows from

and that

mn n mn n

mn 2n1 1

D T S S Z S n T D

T S S

mn n

Thus Z is obtained by removing the equivariant handle T D

mn n

S from Z S and then attaching T S D along the b oundary

mn n

T S S

As we mentioned ab ove the connected sum with the b oundary of simplex is

a bistellar move Other bistellar moves also can b e interpreted as equivariant

surgery op erations on Z

Construction equivariant surgery op erations Let K b e an n

dimensional pure simplicial complex on m and let K b e an n k simplex

6 k 6 n such that link is the b oundary of a k simplex that is not a

face of K Let K b e the complex obtained from K by applying the corresp onding

bistellar k move see Denition

K K n

note that due to our assumptions K has the same numb er of vertices as K

The momentangle complexes corresp onding to and are D

nk k

S and S D resp ectively this follows from Example and

Construction Using stabilization arguments Prop osition we obtain

mn nk k mn nk k

Z Z n T D S T S D

K K

nk k

mn mn

where T S D is attached along its b oundary T

nk k

S S This describ es the b ehavior of Z under bistellar k moves the

cases k and k n are covered by

MOMENTANGLE COMPLEXES

Lemma Let K K be a simplicial sphere and K the simplicial

sphere obtained from K by applying a bistel lar k move k n

Then the corresponding momentangle manifolds Z and Z are T equivariantly

cobordant If K is obtained from K by applying a move then Z is cobordant

to Z S

Proof We give a pro of for k moves k The case k is considered

similarly Consider the pro duct U Z of Z with a segment Dene

K K

nk nk k

mn k mn

X T D S and Y T D D the latter

is a solid equivariant handle Since X Z and X Y we can attach Y to

U at X Z Denote the resulting manifold with b oundary by V ie

Y Then it follows from that V Z Z here Z comes V U

K K K X

from Z U while Z is replaced by Z This concludes the pro of

K K K

Now we have the following top ological corollary of Pachners Theorem

Theorem Let K be a PL sphere Then for some p the momentangle

p n mpn

manifold Z T is equivariantly cobordant to S T This cobordism

is realized by a sequence of equivariant surgeries

Proof By Theorem the PL sphere K is taken to by a sequence of

bistellar moves Since Z S the statement follows from Lemma

Borel constructions and DavisJanuszkiewicz space

Here we study basic homotopy prop erties of Z We also provide necessary

arguments for the statements ab out the cohomology of quasitoric manifolds which

we left unproved in section

m m

Let ET b e the contractible space of the universal principal T bundle over

m m

classifying space BT It is well known that BT is homotopy equivalent to the

pro duct of m copies of innitedimensional pro jective space C P The cell decom

p osition of C P with one cell in every even dimension determines the canonical

m m

cell decomp osition of BT The cohomology of BT with co ecients in k is

thus the p olynomial ring kv v deg v

m i

Definition Let X b e a T space The Borel construction alterna

tively homotopy quotient or associated bund le is the identication space

m m

ET X ET X

m m

where e x eg g x for any e ET x X g T

The pro jection e x e displays ET X as the total space of a bundle

m m m

ET X BT with bre X and structure group T At the same time

m m m

there is a principal T bundle ET X ET X

In the sequel we denote the Borel construction ET X corresp onding to

a T space X by B X In particular for any simplicial complex K on m vertices

we have the Borel construction B Z and the bundle p B Z BT with

T K T K

bre Z

m m

For each i m denote by BT the ith factor in BT C P For

a subset m we denote by BT the pro duct of BT s with i Obviously

m k

BT is a cellular sub complex of BT and BT BT if j j k

BOREL CONSTRUCTIONS AND DAVISJANUSZKIEWICZ SPACE

Definition Let K b e a simplicial complex We refer to the cellular

sub complex

BT BT

as the DavisJanuszkiewicz space and denote it DJ K

The following statement is an immediate corollary of the denition of Stanley

Reisner ring kK Denition

Proposition The cel lular cochain algebra C DJ K and the cohomol

ogy algebra H DJ K are isomorphic to the face ring kK The cel lular inclusion

i DJ K BT induces the quotient epimorphism i kv v kK

kv v I in the cohomology

m K

Theorem The bration p B Z BT is homotopy equivalent to

T K

the cel lular inclusion i DJ K BT More precisely there is a deformation

retraction B Z DJ K such that the diagram

T K

B Z BT

T K

DJ K BT

is commutative

Proof Consider the decomp osition Since each B Z is T stable

the Borel construction B Z ET Z is patched from the Borel construc

T K T K

m j mj

tions ET B for K Supp ose j j j then B D T see

m j j mj

By the denition of Borel construction ET B ET j D ET

j j j j

The space ET D is the total space of a D bundle over BT It follows

that there is a deformation retraction ET B BT which denes a homo

m m

topy equivalence b etween the restriction of p B Z BT to ET B and

T K T

the cellular inclusion BT BT These homotopy equivalences corresp onding

to dierent simplices K t together to yield a required homotopy equivalence

m m

b etween p B Z BT and i DJ K BT

T K

Corollary The momentangle complex Z is the homotopy bre of the

cel lular inclusion i DJ K BT

As a corollary we get the following statement rstly proved in Theo

rem

Corollary The cohomology algebra H B Z is isomorphic to the

T K

face ring kK The projection p B Z BT induces the quotient epimor

T K

phism p kv v kK kv v I in the cohomology

m m K

Corollary The T equivariant cohomology of Z is isomorphic to the

StanleyReisner ring of K

kK H Z

The following information ab out the homotopy groups of Z can b e retrieved

from the ab ove constructions

MOMENTANGLE COMPLEXES

Theorem a The complex Z is connected ie Z Z

K K K

and Z B Z DJ K for i >

i K i T K i

b If K K and P is q neighborly see Denition then Z

P i K

for i q Moreover Z is a free Abelian group generated by the

q P

q element missing faces of K

m m

Proof Note that BT K Z and the skeleton of DJ K coincides

with that of BT If P is q neighb orly then it follows from Denition that the

q skeleton of DJ K coincides with that of BT Now b oth statements

follow easily from the exact homotopy sequence of the map i DJ K BT with

homotopy bre Z see Corollary

Remark We say that a simplicial complex K on the set m is k neighborly

if any k element subset of m is a simplex of K This denition is an obvious

extension of the notion of k neighb orly simplicial p olytop e to arbitrary simplicial

complexes Then the second part of Theorem holds for arbitrary q neighb orly

simplicial complex

Supp ose now that K K for some simple np olytop e P and M is a qua

sitoric manifold over P with characteristic function see Denition Then we

m mn

have the subgroup H T acting freely on Z and the principal T bundle

Z M Prop osition

n n

Proposition The Borel construction ET M is homotopy equiv

alent to B Z

T P

Proof Since H acts freely on Z we have

B Z ET Z

T P T P

m n n

EH E T H Z H ET M

P T

T H

Corollary The T equivariant cohomology ring of a quasitoric mani

n n n

fold M over P is isomorphic to the StanleyReisner ring of P

n n

M kP H

Proof It follows from Prop osition and Corollary

Theorem Theorem The LeraySerre spectral sequence of the

bund le

n n n

ET M BT

n pq

with bre M col lapses at the E term ie E E

n n

Proof Since b oth BT and M have only evendimensional cells see Prop o

sition all the dierentials in the sp ectral sequence are trivial by dimensional

reasons

Corollary Projection induces a monomorphism kt t

n n n

kP in the cohomology The inclusion of bre M ET M induces an

epimorphism kP H M

Now we are ready to give pro ofs for the statements from section

WALK AROUND THE CONSTRUCTION OF Z

Proof of Lemma and Theorem The monomorphism

n n n

H BT kt t kP H ET M

n T

takes t to i n By Theorem kP is a free kt t mo dule

i i n

hence is a regular sequence Therefore the kernel of kP H M

is exactly J

Walk around the construction of Z generalizations analogues

and additional comments

Many of our previous constructions namely the cubical complex ccK the

momentangle complex Z the Borel construction B Z the DavisJanuszkiewicz

K T K

space DJ K and also the complement U K of a co ordinate subspace arrangement

app earing in section admit a unifying combinatorial interpretation in terms of

the following construction which was mentioned to us by N Strickland in private

communications

Construction Let X b e a space and W a subspace of X Let K b e a

simplicial complex on the set m Dene the following subset in the pro duct of m

copies of X

Y Y

X W X W K

Example ccK K I see

Z K D S see

DJ K K C P see Denition

1 1

B Z K ES D E S S see the pro of of Theorem

T K

S S

Another unifying description of the ab ove spaces can b e achieved using categor

ical constructions of limits and colimits of dierent diagrams over the face category

catK of K The ob jects of catK are simplices K and the morphisms

are inclusions For instance Denition is an example of this pro cedure the

DavisJanuszkiewicz space is the colimit of the diagram of spaces over catK that

assigns BT to a simplex K In the case when K is a ag complex see Def

inition and Prop osition the colimit over catK reduces to the graph

product studied in the theory of groups see eg Wellknown examples of

graph pro ducts include rightangled Coxeter and Artin groups see eg

The most general categorical setup for the ab ove constructions involves the notion

of homotopy colimit This fundamental algebraictop ological concept

has already found combinatorial applications see The complex Z can b e

seen as the homotopy colimit of a certain diagram of tori this interpretation is

similar to the homotopy colimit description of toric varieties prop osed in

For more information on this approach see

The combinatorial theory of toric spaces is parallel to some extent to its Z

or real counterpart We say a few words ab out the Ztheory here referring

the reader to and other pap ers of R Charney M Davis T Januszkiewicz

and their coauthors for a more detailed treatment some further results can b e also

found in The rst step is to pass from the torus T to its real analogue

m m m

the group Z The standard cub e I is the orbit space for the

m m

action of Z on the bigger cub e which in turn can b e regarded as a

m m

real analogue of the p olydisc D C Now given a cubical sub complex

MOMENTANGLE COMPLEXES

m m

C I one can construct a Z symmetrical cubical complex emb edded into

just in the same way as it is done in Denition In particular for any

simplicial complex K on the vertex set m one can intro duce the real versions RZ

and RW of the momentangle complexes Z and W In the notations of

K K K

Construction we have

RZ K f g

This cubical complex was studied eg in under the name mirroring construc

tion If K is a simplicial n sphere then RZ is an ndimensional manifold the

pro of is similar to that of Lemma Thereby for any simplicial sphere K

m m

with m vertices we get a Z symmetric nmanifold with a Z invariant cu

bical sub division As suggested by the results of this class of cubical manifolds

may b e useful in the combinatorial theory of face vectors of cubical complexes see

section The real analogue RZ of the manifold Z corresp onding to the case

P P

of a p olytopal simplicial sphere is the universal Abelian cover of the p olytop e P

regarded as an orbifold or manifold with corners see eg x In man

ifolds RZ and Z are interpreted as the conguration spaces of equivariant hinge

P P

mechanisms or linkages in R and R

Example Let P b e an mgon Then RZ is a dimensional manifold

2 2

It is easy to see that RZ RZ S a sphere patched from triangles

1 1 2

RZ T a torus patched from squares More generally and RZ

RZ is patched from p olygons meeting by at each vertex Hence we have

m m

m vertices and m edges so the Euler characteristic is

RZ m

m m

Thus RZ is a surface of genus m This also can b e seen directly

into a connected sum of an m gon and triangle and using by decomp osing P

the real version of Example

n n

Replacing T by Z in Denition we obtain a real version of quasitoric

manifolds which was intro duced in under the name smal l covers Thereby a

n n n n

small cover of a simple p olytop e P is a Z manifold M with quotient P

The name refers to the fact that any branched cover of P as an orbifold by a

smo oth manifold has at least sheets Small covers were studied in along with

quasitoric manifolds and many results on quasitoric manifolds quoted from in

section have analogues in the small cover case Also like in the torus case

every small cover is the quotient of the universal cover RZ by a free action of the

group Z

An imp ortant class of small covers and quasitoric manifolds was intro duced

in Example under the name pul lbacks from the linear model They cor

resp ond to simple p olytop es P whose dual triangulation can b e folded onto the

n simplex more precisely the p olytopal sphere K admits a nondegenerate

simplicial map onto note that this is always the case when K is a barycen

tric sub division of some other p olytopal sphere see Example If this condition

is satised then there exists a sp ecial characteristic map which assigns to each

n n n

facet of P a co ordinate subtorus T T or co ordinate subgroup Z Z

i i

in the small cover case Pullbacks from the linear mo del have a numb er of nice

prop erties in particular they are all stably parallelizable Corollary

compare with Theorem The existence of a nondegenerate simplicial map

WALK AROUND THE CONSTRUCTION OF Z

n n

from K to can b e reformulated by saying that the p olytop e P admits a

regular npaint coloring The latter means that the facets of P can b e colored

with n paints in such a way that any two adjacent facets have dierent color A

simple p olytop e P admits a regular npaint coloring if and only if every face

has an even numb er of edges This is a classical result for n the pro of in the

general case can b e found in Some additional results ab out pullbacks from

the linear mo del in dimension were obtained in It was shown there that any

small cover M which is a pullback from the linear mo del admits an equivariant

emb edding into R R R with the standard action of Z on R and the

trivial action on R Another result from says that any such M can b e ob

tained from a set of dimensional tori by applying several equivariant connected

sums and equivariant Dehn twists compare with section

Although not every simple p olytop e admits a regular paint coloring a reg

ular paint coloring can always b e achieved due to the Four Color Theorem This

argument was used in Example to prove that there is a small cover or

quasitoric manifold over every simple p olytop e One can just construct a char

acteristic map by assigning the co ordinate circles in T to the rst three colors and

the diagonal circle to the fourth one

On the other hand it would b e particularly interesting to develop a quaternionic

analogue of the theory Unlike the real case not much is done here To b egin of

n n n

course we have to replace T by the quaternionic torus S p S Developing

quaternionic analogues of toric and quasitoric manifolds is quite tricky R Scott

in used the quaternionic analogue of characteristic map to approach this

problem However the noncommutativity of the quaternionic torus implies that

it do es not contain suciently many subgroups for the resulting quaternionic toric

manifolds to have an actual S p action A p olytopal structure also app ears in the

quotients of some other typ es of manifolds studied in the quaternionic geometry see

eg We also mention that since only co ordinate subgroups of T are involved

in the denition of the momentangle complex Z this particular construction of

a toric space do es have a quaternionic analogue which is an S p space

At the end we give one example which builds on a generalization of the con

struction of Z to the case of an arbitrary group G

Example classifying space for group G Let K b e a simplicial complex

on the vertex set m Set Z G K coneG G see Construction where

coneG is the cone over G with the obvious Gaction By the construction the

group G acts on Z G with quotient coneK It is also easy to observe that

the diagonal subgroup in G acts freely on Z G thus identifying Z G as a

K K

principal Gspace

Supp ose now that K K K is a sequence of emb edded

simplicial complexes such that K is ineighb orly The group G acts freely on the

contractible space lim Z G and the corresp onding quotient is thus the classifying

space BG Thus we have the following ltration in the universal bration EG

Z G Z G Z G

K K K

1 2 i

Z GG Z GG Z GG

K K K

1 2 i

The wellknown Milnor ltration in the universal bration of the group G corre

sp onds to the case K i

MOMENTANGLE COMPLEXES

CHAPTER

Cohomology of momentangle complexes and

combinatorics of triangulated manifolds

The Eilenb ergMo ore sp ectral sequence

In their pap er of Eilenb erg and Mo ore constructed a sp ectral se

quence of great imp ortance for algebraic top ology This sp ectral sequence can b e

considered as an extension of Adams approach to calculating the cohomology of

lo op spaces In s dierent applications of the Eilenb ergMo ore sp ec

tral sequence led to many imp ortant results on the cohomology of lo op spaces and

homogeneous spaces for Lie group actions In this chapter we discuss some new

applications of this sp ectral sequence to combinatorial problems This section con

tains the necessary information ab out the sp ectral sequence we follow L Smiths

pap er in this description

The following theorem provides an algebraic setup for the Eilenb ergMo ore

sp ectral sequence

Theorem Eilenb ergMo ore Theorem Let A be a commuta

tive dierential graded kalgebra and M N dierential graded Amodules Then

there exists a spectral sequence fE d g converging to Tor M N and whose E

r r A

term is

ij ij

H M H N i j > Tor E

H A

where H denotes the cohomology algebra or module

The ab ove sp ectral sequence lives in the second quadrant and its dierentials

d add r r to bidegree r > It is called the algebraic EilenbergMoore

spectral sequence For the corresp onding decreasing ltration fF Tor M N g

in Tor M N we have

X X

ij ij

p pnp p

F E M N M N F Tor Tor

A A

ij n ij n

Top ological applications of Theorem arise in the case when A M N are

singular or cellular co chain algebras of certain top ological spaces The classical

situation is describ ed by the commutative diagram

E E

y y

B B

where E B is a Serre bre bundle with bre F over a simply connected base B

and E B is the pullback along a continuous map B B For any space X let

C X denote either the singular co chain algebra of X or in the case when X is a

COHOMOLOGY OF MOMENTANGLE COMPLEXES

cellular complex the cellular co chain algebra of X Obviously C E and C B

are C B mo dules Under these assumptions the following statement holds

Lemma Prop osition Tor C E C B is an algebra

C B

in a natural way and there is a canonical isomorphism of algebras

Tor C E C B H E

C B

Applying Theorem in the case A C B M C E N C B and

taking into account Lemma we come to the following statement

Theorem Eilenb ergMo ore There exists a spectral sequence of commu

tative algebras fE d g with

r r

a E H E

ij ij

b E Tor H E H B

H B

The sp ectral sequence of Theorem is called the top ological Eilenberg

Moore spectral sequence The case when B in is a p oint is particularly imp or

tant for applications so we state the corresp onding result separately

Corollary Let E B be a bration over a simply connected space B

with bre F Then there exists a spectral sequence of commutative algebras fE d g

r r

with

a E H F

b E Tor H E k

H B

We refer to the sp ectral sequence of Corollary as the EilenbergMoore spec

tral sequence of bration E B

n n

Example Let M b e a quasitoric manifold over P see Denition

n n

Consider the Eilenb ergMo ore sp ectral sequence of the bundle ET M

n n n n

BT with bre M By Prop osition H ET M H B Z

T T P

kP The monomorphism

n n n n

kt t H BT H ET M kP

n T

takes t to i n see The E term of the Eilenb ergMo ore sp ectral

i i

sequence is

n n n

kP k H ET M k Tor Tor E

kt t H BT

1 n

Since kP is a free kt t mo dule we have

n n

kP k kP k Tor Tor

kt t kt t

1 n 1 n

n n

kP k kP

kt t

1 n

Therefore E kP J and E for p It follows that the Eilenb erg

n n

Mo ore sp ectral sequence collapses at the E term and H M kP J in

accordance with Theorem

Cohomology algebra of Z

Here we apply the Eilenb ergMo ore sp ectral sequence to calculating the coho

mology algebra of the momentangle complex Z As an immediate corollary we

obtain that the cohomology algebra inherits a canonical bi grading from the sp ec

tral sequence The corresp onding bigraded Betti numb ers coincide with imp ortant

combinatorial invariants of K intro duced by Stanley

COHOMOLOGY ALGEBRA OF Z

Theorem The fol lowing isomorphism of algebras holds

kK k H Z Tor

kv v

1 m

This formula either can be seen as an isomorphism of graded algebras where the

grading in the right hand side is by the total degree or used to dene a bigraded

algebra structure in the left hand side In particular

H Z Tor kK k

kv v

1 m

ij p

Proof Let us consider the Eilenb ergMo ore sp ectral sequence of the commu

tative square

E ET

y y

DJ K BT

where the left vertical arrow is the pullback along i Corollary shows that E

is homotopy equivalent to Z

By Prop osition the map i DJ K BT induces the quotient epimor

phism

i C BT kv v kK C DJ K

where C denotes the cellular co chain algebra Since ET is contractible there

m m

is a chain equivalence C ET k More precisely C ET can b e identied

with the Koszul resolution u u kv v of k see Example

m m

Therefore we have an isomorphism

DJ K C ET Tor kK k Tor C

kv v C BT

1 m

The Eilenb ergMo ore sp ectral sequence of commutative square has

DJ K H ET E Tor H

H BT

and converges to Tor C DJ K C ET Theorem Since

C BT

DJ K H ET Tor kK k Tor H

kv v H BT

1 m

it follows from that the sp ectral sequence collapses at the E term that is

E E Lemma shows that the mo dule Tor C DJ K C ET

C BT

is an algebra isomorphic to H Z which concludes the pro of

Theorem displays the cohomology of Z as a bi graded algebra and says that

the corresp onding bigraded Betti numb ers b Z coincide with that of kK

see The next theorem follows from Lemma and Corollary

Theorem The fol lowing isomorphism of bigraded algebras holds

H u u kK d H Z

m K

where the bigraded structure and the dierential in the right hand side are dened

COHOMOLOGY OF MOMENTANGLE COMPLEXES

In the sequel given two subsets fi i g fj j g of m we

p q

will denote the squarefree monomial

u u v v u u kK

i i j j m

1 p 1 q

by u v Note that bideg u v p p q

Remark Since the dierential d in do es not change the second degree

the dierential bigraded algebra u u kK d splits into the sum of

dierential subalgebras consisting of elements of xed second degree

Corollary The LeraySerre spectral sequence of the principal T bund le

ET Z B Z col lapses at the E term

K T K

Proof The sp ectral sequence under consideration converges to H ET

Z H Z and has

K K

E H T H B Z u u kK

T K m

The dierential in the E term acts as in Hence

E H E d H u u kK H Z

m K

by Theorem

Construction Consider the subspace A K u u kK

spanned by monomials u and u v such that is a simplex of K j j q and

Dene

A K A K

q q

Since du v and dv we have dA K A K Therefore A K

i i i

is a co chain sub complex in u u kK d Moreover A K inherits the

bigraded mo dule structure from u u kK with dierential d adding

to bidegree Hence we have an additive inclusion ie a monomorphism of

bigraded mo dules i A K u u kK On the other hand A K

a m

is an algebra in the obvious way but is not a subalgebra of u u kK

For instance v in A K but v in u u kK Never

theless we have multiplicative projection an epimorphism of bigraded algebras

j u u kK A K The additive inclusion i and the multiplica

m m a

tive pro jection j satisfy j i id

m m a

Lemma The cochain complexes u u kK d and A K d

are cochain homotopy equivalent and therefore have the same cohomology This

implies the fol lowing isomorphism of bigraded kmodules

Tor H A K d kK k

kv v

1 m

for Proof A routine check shows that the co chain homotopy op erator s

the Koszul resolution see the pro of of Prop osition VI I in establishes a

co chain homotopy equivalence b etween the maps id and i j from the algebra

a m

u u kK d to itself That is

ds sd id i j

a m

COHOMOLOGY ALGEBRA OF Z

We just illustrate the ab ove identity on few simple examples

su v u u dsu v u v u v sdu v u v u v

hence ds sdu v id i j u v

a m

su v u dsu v du v v sdu v u v

hence ds sdu v u v id i j u v

a m

sv u v dsv v dv

hence ds sdv v id i j v

a m

Now we recall our cell decomp osition of Z see Lemma The cells of

Z are the sign vectors T fD T g with T K Assign to each pair of

K D

disjoint subsets of m the vector T with T T Then

D T

T is a cell of Z if and only if K Let C Z and C Z denote the

K K K

cellular chain and co chain complexes of Z resp ectively Both complexes C Z

K K

and A K have the same cohomology H Z The complex C Z has the

K K

canonical additive basis consisting of co chains T As an algebra C Z is

of dimension and resp ectively dual to the T generated by the co chains D

j i

cells D T fig and T T fj g 6 i j 6 m At the same time A K

i j

is multiplicatively generated by v u 6 i j 6 m

i j

Theorem The correspondence v u T establishes a canonical

isomorphism between the dierential graded algebras A K and C Z

Proof It follows directly from the denitions of A K and C Z that the

prop osed map is an isomorphism of graded algebras So it remains to prove that

it commutes with dierentials Let d d and denote the dierential in A K

c c

C Z and C Z resp ectively Since dv and du v we need to show

K K i i i

We have D T T A cell of Z D d T that d D

c i i c i K c c

i i i

is either D or T T T k j Then

j j k j k

T i D i hT D i hT T i hd T T i hT hd T

j c j j c j k c j k ij c

i i i i i

Further a cell D where if i j and otherwise Hence d T

ij ij c

i i

of Z is either D T or T T T T Then

K j k j j j j j j

1 2 3 1 2 3

T i D T i hD D T i hD hd D

j k c j k j k c

i i i

T i T i hD hd D

c j j j j j j c

i 1 2 3 i 1 2 3

Hence d D

The ab ove theorem provides a top ological interpretation for the dierential algebra

A K d In the sequel we will not distinguish the co chain complexes A K and

C Z and identify u with T v with D

K i i

i i

Now we can summarise the results of Prop osition Lemma Lemma

Corollary and Theorem in the following statement describing the functorial

prop erties of the corresp ondence K Z

Proposition Let us introduce the fol lowing functors

Z the covariant functor K Z from the category of nite simplicial

complexes and simplicial inclusions to the category of toric spaces and equi

variant maps the momentangle complex functor

COHOMOLOGY OF MOMENTANGLE COMPLEXES

k the contravariant functor K kK from simplicial complexes to

graded kalgebras the StanleyReisner functor

Toralg the contravariant functor

kK k K Tor

kv v

1 m

from simplicial complexes to bigraded kalgebras the Toralgebra functor it

coincides with the composition of k and Tor k

kv v

1 m

H the contravariant functor X H X from the category of toric spaces

T T

and equivariant maps to kalgebras the equivariant cohomology functor

H the contravariant functor X H X from spaces to kalgebras the

ordinary cohomology functor

Then we have the fol lowing identities

H Z k H Z Toralg

The later identity implies that for every simplicial inclusion K K the

cohomology map H Z H Z coincides with the induced homo

K K

2 1

morphism of Toralgebras In particular induces a homomorphism

q p q p

H Z H Z of bigraded cohomology modules

K K

2 1

In the CohenMacaulay case we have the following reduction theorem for the

cohomology of Z

Theorem Suppose that K is CohenMacaulay and let J be an ideal

in kK generated by degreetwo regular sequence of length n Then the fol lowing

isomorphism of algebras holds

H Z Tor kK J k

kv v J

1 m

Proof This follows from Theorem and Lemma

Note that the kalgebra kK J is nitedimensional unlike kK In some cir

cumstances see section this helps to calculate the cohomology of Z more

eciently

Bigraded Betti numb ers of Z the case of general K

The bigraded structure in the algebra A K d denes a bigrading in the

cellular chain complex C Z via the isomorphism of Theorem We have

K c

bideg D bideg T bideg

i i i

The dierential adds to bidegree and thus the bigrading descends to the

cellular homology of Z

In this section we assume that the ground eld k is of zero characteristic Dene

the bigraded Betti numb ers

b Z dim H C Z q p m

q p K q p K c

Theorem and Lemma show that

q p

b Z dim Tor kK k kK

q p K

kv v

1 m

BIGRADED BETTI NUMBERS OF Z THE CASE OF GENERAL K

see Alternatively b Z equals the dimension of q pth bigraded

q p K

comp onent of the cohomology algebra H u u kK d For the ordinary

Betti numb ers b Z we have

k K

b Z b Z k m n

k K q p K

q pk

The lemma b elow describ es some basic prop erties of bigraded Betti num

b ers

Lemma Let K be a simplicial complex with m f vertices and f

edges and Z the corresponding momentangle complex dim Z m n Then

K K

a b Z b Z and b Z for p

K K p K

b b for p m or q p

q p

c b Z b Z

K K

d b Z b Z f

K K

e b Z for q > p or p q n

q p K

f b Z b Z

mn K K

mnm

Proof In this pro of we calculate the Betti numb ers using the co chain sub

complex A K u u kK The mo dule A K has the basis con

sisting of monomials u v with K and Since bideg v

q p

bideg u the bigraded comp onent A K is spanned by monomials

q p

u v with j j p q and j j q In particular A K if p m or q p

whence the assertion b follows To prove a we observe that A K is generated

p p

by while any v A K p is a cob oundary whence H Z for

q p

Now lo ok at assertion e Every u v A K has K while any

q p

simplex of K has at most n vertices It follows that A K for p q n

By b b Z for q p so it remains to prove that b Z

q p K q q K

q q

for q The mo dule A K is generated by monomials u with j j q

q q

Since du v it follows easily that there are no nonzero co cycles in A K

i i

q q

Hence H Z

The assertion c follows from e and

It also follows from e that H Z H Z The basis for A K

K K

consists of monomials u v i j We have du v v v and du u u v

j i j i i j i j j i

u v Hence u v is a co cycle if and only if fi j g is not a simplex in K in this

i j j i

case two co cycles u v and u v represent the same cohomology class Assertion d

j i i j

follows

The remaining assertion f follows from the fact that a monomial u v

A K has maximal total degree m n if and only if j j n and j j m n

Lemma shows that nonzero bigraded Betti numb ers b Z r

rp K

app ear only in the strip b ounded by the lines p m r p r and

p r n in the second quadrant see Figure a

The homogeneous comp onent C Z has basis of cellular chains T

q p K

with K j j p q and j j q It follows that

mpq

dim C Z f

q p K pq q

COHOMOLOGY OF MOMENTANGLE COMPLEXES

m m

m n m n

n n

b jK j S a arbitrary K

Figure Possible lo cations of nonzero bigraded Betti num

b ers b Z marked by

q p K

where f f f is the f vector of K and f The dierential

n c

do es not change the second degree

C Z C Z

c q p K q p K

Hence the chain complex C Z splits as follows

C Z C Z

K c p K c

Remark The similar decomp osition holds also for the cellular co chain com

plex C Z d A K d

K c

Let us consider the Euler characteristic of complex C Z

p K c

m m

X X

q q

Z dim C Z b Z

p K q p K q p K

q q

Dene the generating p olynomial Z t by

Z t Z t

K p K

The following theorem calculates this p olynomial in terms of the hvector of K

Theorem For every n dimensional simplicial complex K with m

vertices it holds that

mn n m

Z t t h h t h t t F kK t

K n

where h h h is the hvector of K

Proof It follows from and that

m j

Z f

p K j

p j

BIGRADED BETTI NUMBERS OF Z THE CASE OF GENERAL K

Then

m m m

X X X

m j

p j pj pj

Z t K t t t f

K p j

p j

p p

n m

X X

j mj m j

f t t t f t

j j

Denote ht h h t h t From we get

n n i

t ht t f t

Substituting t for t ab ove we nally rewrite as

Z t t ht ht

m n n

t t t

which is equivalent to the rst identity from The second identity follows

from Lemma

The formula from the ab ove theorem can b e used to express the face vector of

a simplicial complex in terms of the bigraded Betti numb ers of the corresp onding

momentangle complex Z

Corollary The Euler characteristic of Z is zero

Proof We have

m m

X X

q p

Z b Z Z Z

K q p K p K K

pq p

so the statement follows from

Remark Another pro of of the ab ove corollary follows from the observation

that the diagonal subgroup S T always acts freely on Z see section

Hence there exists a principal S bundle Z Z S which implies Z

K K K

The torus Z T is a cellular sub complex of Z see

Lemma The cellular co chain sub complex C Z C Z A K has

the basis consisting of co chains T and is mapp ed to the exterior algebra

u u A K under the isomorphism of Theorem It follows that

there is an isomorphism of kmo dules

C Z Z A K u u

K m

We intro duce relative bigraded Betti numb ers

q p

b Z Z dim H C Z Z d q p m

q p K K

dene the pth relative Euler characteristic Z Z by

p K

m m

X X

q q p q

Z Z dim C Z Z b Z Z

p K K q p K

q q

and dene the corresp onding generating p olynomial

Z Z t Z Z t

K p K

COHOMOLOGY OF MOMENTANGLE COMPLEXES

Theorem For any n dimensional simplicial complex K with m

vertices it holds that

mn n m

Z Z t t h h t h t t

K n

Proof Since C Z u u and bideg u we have

m i

q q q

dim C Z dim C Z

Combining and we get

p pp

Z Z Z dim C Z

p K p K

Hence

t Z Z t Z t

K K

mn n m

t h h t h t t

We will use the ab ove theorem in section

Bigraded Betti numb ers of Z the case of spherical K

If K is a simplicial sphere then Z is a manifold Lemma This im

p oses additional conditions on the cohomology of Z and leads to some interesting

interpretations of combinatorial results and problems from chapters and

Theorem Let K be an n dimensional simplicial sphere and Z

the corresponding momentangle manifold dim Z m n Then the fundamental

cohomology class of Z is represented by any monomial v u A K of bidegree

m n m such that is an n simplex of K and The sign

depends on a choice of orientation for Z

mn mnm

Proof Lemma f shows that H Z H Z The

K K

mnm n

mo dule A K is spanned by the monomials v u such that K

j j n m n Every such monomial is a co cycle Supp ose that are

two n simplices of K sharing a common n face We claim that the

0 0

corresp onding co cycles v u v u where m n m n represent

the same cohomology class up to a sign Indeed let

v u v v u u

i i j j

1 n 1 mn

0 0

v u v v v u u u

i i j i j j

1 n1 1 n 2 mn

Since every n face of K is contained in exactly two n faces the identity

dv v u u u u

i i i j j j

1 n1 n 1 2 mn

v v u u v v v u u u

i i j j i i j i j j

1 n 1 mn 1 n1 1 n 2 mn

0 0

holds in A K u u kK Hence v u v u as cohomology

classes Since K is a simplicial sphere every two n simplices can b e

connected by a chain of simplices in such a way that any two successive simplices

mnm

share a common n face Thus all monomials v u in A K represent

the same cohomology class up to a sign This class is a generator of H Z

ie the fundamental cohomology class of Z K

BIGRADED BETTI NUMBERS OF Z THE CASE OF SPHERICAL K

Remark In the ab ove pro of we have used two combinatorial prop erties of

K The rst one is that every n face is contained in exactly two n

faces and the second is that every two n simplices can b e connected by a

chain of simplices with any two successive simplices sharing a common n face

Simplicial complexes satisfying these two conditions are called pseudomanifolds

In particular every triangulated manifold is a pseudomanifold Hence for any

triangulated manifold K we have b Z b Z and the

mn K K

mnm

generator of H Z can b e chosen as describ ed in Theorem

Corollary The Poincare duality for the moment angle manifold Z

corresponding to a simplicial sphere K respects the bigraded structure in the

cohomology ie

q p

H Z H Z

K K

mnq mp

In particular

b Z b Z

q p K K

mnq mp

Corollary Let K be an n dimensional simplicial sphere and

Z the corresponding momentangle complex dim Z m n Then

K K

a b Z for q > m n with only exception b

q p K

mnm

b b Z for p q > n with only exception b

q p K

mnm

It follows that if K is a simplicial sphere then nonzero bigraded Betti

numb ers b Z with r and r m n app ear only in the strip b ounded by

rp K

the lines r m n r p r and p r n in the second

quadrant see Figure b Compare this with Figure a corresp onding to

the case of general K

Example Let K Then kK kv v v v

m m

see Example A direct calculation shows that the cohomology H kK

u u d see Theorem is additively generated by the classes and

v v v u We have deg v v v u m and Theorem says

m m m m

that v v v u represents the fundamental cohomology class of Z S

m m K

Example Let K b e the b oundary complex of an mgon P with m >

We have kK kv v I where I is generated by the monomials v v

m P P i j

i j mo d m The complex Z Z is a manifold of dimension m The

K P

Betti numb ers of these manifolds were calculated in Namely

for k m

for k m m

dim H Z

m m m

m for 6 k 6 m

k k k

For example in the case m the group H Z has generators represented

by the co cycles v u kK u u i while the group

i i

H Z has generators represented by the co cycles v u u j

P j j j

As it follows from Theorem the pro duct of co cycles v u and v u u

i i j j j

represents a nonzero cohomology class in H Z if and only if all the indices

i i j j j are dierent Thus for each of the cohomology classes

v u there is a unique Poincare dual cohomology class v u u such that

i i j j j

the pro duct v u v u u is nonzero This observation has the following

i i j j j

COHOMOLOGY OF MOMENTANGLE COMPLEXES

generalization which describ es the multiplicative structure in the cohomology of

Z for any mgon P

Let P be an mgon m > Proposition Cohomology ring of Z

a The only nonzero bigraded cohomology groups of Z are H H

pp p mm m

H H for p m and H H

b The group H is free and is generated by the cohomology classes

v u such that j j p i and i These cohomology classes are

subject to relations of the form du for j j p The corresponding

Betti numbers are given by

c The group H is onedimensional with generator v v u u

p p

1 1

d The product of two cohomology classes v u H and v u

i i

1 1 2 2

p p

2 2

H equals v v u u up to a sign if ffi g fi g g is a

partition of m and zero otherwise

Therefore the only nontrivial products in the ring H Z are those which give

a multiple of the fundamental class

Proof Statement a follows from Corollary b is obvious and c

follows from Theorem In order to prove d we mention that the pro duct of two

p p p p

1 1 2 2

classes a H and a H has bidegree p q with q p

whence it can b e nonzero only if it b elongs to H by Corollary

It follows from and that for any simplicial sphere K the following

holds

Z Z

p K mp K

From this and we get

m n

t t h h t h t

m m n

n m

t t

m n

t t h h t h t

m n

n n

m n

t t

n n

h t h t h

Hence h h Thus the DehnSommerville equations are a corollary of the

i ni

bigraded Poincare duality

The identity also allows us to interpret dierent inequalities for the f

vectors of simplicial spheres or triangulated manifolds in terms of top ological invari

ants the bigraded Betti numb ers of the corresp onding momentangle manifolds

or complexes Z

Example Using the expansion

m n i

t i

together with the UBC for simplicial spheres Corollary and identity

we deduce that the inequality Z t 6 holds co ecientwise for any simplicial

sphere K That is

Z t 6 for i

i K

PARTIAL QUOTIENTS OF Z

Example Using Lemma we calculate

Z Z

K K

Z b Z b Z Z b Z b Z

K K K K K K

note that b Z b Z and b Z b Z b Z Now

K K K K K

identity shows that

h m n

b Z h

h m nb Z b Z b Z

K K K

It follows that the inequality h 6 h n > from the GLBC for simplicial

spheres is equivalent to the following

b Z 6

Note that this inequality is not valid for n see eg Example and b ecomes

identity for n The next inequality h 6 h n > from is equivalent

to the following

m n b Z b Z b Z >

K K K

We see that the combinatorial GLBC inequalities are interpreted as top ologi

cal inequalities for the bigraded Betti numb ers of a manifold This might op en

a p ossibility to use top ological metho ds such as the equivariant top ology or Morse

theory for proving inequalities like or Such a top ological approach

to problems like g conjecture or GLBC has an advantage of b eing indep endent

on whether the simplicial sphere K is p olytopal or not Indeed as we have al

ready mentioned all known pro ofs for the necessity condition in the g theorem for

simplicial p olytop es including the original one by Stanley given in section Mc

Mullens pro of and the recent pro of by Timorin follow the same scheme

Namely the numb ers h i n are interpreted as the dimensions of graded

comp onents A of a certain algebra A satisfying the Hard Lefschetz Theorem The

latter means that there is an element A such that the multiplication by de

n n

i i

nes a monomorphism A A for i This implies h 6 h for i

i i

see section However such an element is lacking for nonp olytopal K which

means that a new technique has to b e develop ed in order to prove the g conjecture

for simplicial spheres

As it was mentioned in section simplicial spheres are Gorenstein com

plexes Using Theorems and our Theorem we obtain the following

answer to a weaker version of Problem

Proposition The complex Z is a Poincare duality complex over k

if and only if K is Gorenstein ie for any simplex K including the

subcomplex link has the homology of a sphere of dimension dim link

artial quotients of Z P

Here we return to the case of p olytopal K ie K K for some simple

p olytop e P and study quotients of Z by freely acting subgroups H T P

COHOMOLOGY OF MOMENTANGLE COMPLEXES

n n

For any combinatorial simple p olytop e P dene s sP to b e the maximal

s m

dimension for which there exists a subgroup H T in T acting freely on Z

n n

The numb er sP is obviously a combinatorial invariant of P

Problem V M Buchstab er Provide an ecient way to calculate the

n n

number sP eg in terms of known combinatorial invariants of P

n n

Proposition If P has m facets then sP 6 m n

Proof Every subtorus of T of dimension m n intersects nontrivially

with any ndimensional isotropy subgroup and therefore cannot act freely on Z

i i

Proposition The diagonal circle subgroup S fe e

m n

T g R acts freely on any Z Thus sP >

Proof By Denition every isotropy subgroup for Z is co ordinate and

therefore intersects S only at the unit

An alternative lower b ound for the numb er sP was prop osed in Let

F fF F g b e the set of facets of P We generalize the denition of a

F k where regular coloring from section as follows A surjective map

k f k g is called a regular k paint coloring of P if F F whenever

i j

F F The chromatic number P is the minimal k for which there exists a

i j

n n

regular k paint coloring of P Then P > n and due to the result mentioned in

section the equality is achieved if and only if every face of P is an evengon

Note also that P 6 by the Four Color Theorem

Example Supp ose P is a neighb orly simple p olytop e with m facets

Then P m

Proposition The fol lowing inequality holds

n n

sP > m P

m k

Proof The map F k denes an epimorphism of tori T T

It is easy to see that if is a regular coloring then Ker T acts freely

on Z

For more results on colorings and their relations with Problem see

Let H T b e a subgroup of dimension r 6 m n Cho osing a basis we can

write it in the form

is s is s m

11 1 1r r m1 1 mr r

H e e T

where R i r The integer m r matrix S s denes a

i ij

r m m

monomorphism Z Z whose image is a direct summand in Z For any subset

fi i g m denote by S the m n r submatrix of S obtained

i i

1 n

by deleting the rows i i Write each vertex v P as an intersection of n

facets as in The following criterion of freeness for the action of H on Z

holds

Lemma Subgroup acts freely on Z if and only if for every vertex

v F F of P the m n r submatrix S denes a monomorphism

i i

1 n

i i

1 n

r mn

Z Z to a direct summand

PARTIAL QUOTIENTS OF Z

Proof It follows from Denition that the orbits of T action on Z cor

resp onding to the vertices of P have maximal rank n isotropy subgroups The

isotropy subgroup corresp onding to a vertex v F F is the co ordinate

i i

1 n

n m

subtorus T T Subgroup acts freely on Z if and only if it inter

i i

1 n

sects each isotropy subgroup only at the unit This is equivalent to the condition

n m

that the map H T T is injective for any v F F This

i i

1 n

i i

1 n

map is given by the integer m n r matrix obtained by adding n columns

with at the place i j n to S The map is

injective if and only if this enlarged matrix denes a direct summand in Z The

latter holds if and only if each S denes a direct summand

i i

1 n

In particular for subgroups of rank m n we get the following statement

Corollary The subgroup of rank r m n acts freely on Z if

and only if for any vertex v F F of P holds det S

i i

1 n

i i

1 n

Proposition A simple polytope P admits a characteristic map if and

only if sP m n

Proof Prop osition shows that if P admits a characteristic map then

the m ndimensional subgroup H acts freely on Z whence sP m n

Now supp ose sP m n ie there exists a subgroup of rank r

m n that acts freely on Z The corresp onding m m nmatrix S denes

mn m

a monomorphism Z Z whose image is a direct summand It follows that

there is an n mmatrix such that the sequence

mn m n

Z Z Z

is exact Since S satises the condition of Corollary the matrix satises

thus dening a characteristic map for P

n n

Supp ose M is a quasitoric manifold over P with characteristic map Write

the subgroup H in the form Now dene the following linear forms in

kv v

w s v s v i m n

i i mi m

Under these assumptions the following statement holds

Lemma There is the fol lowing isomorphism of algebras

H Z Tor H M k

kw w

1 mn

where the kw w module structure in H M kv v I J

mn m P

is dened by

Proof By Theorem

Tor H Z kK J k

kv v J

1 m

The quotient kv v J is identied with kw w

m mn

Theorem The LeraySerre spectral sequence of the T bund le Z

M col lapses at the E term Furthermore the fol lowing isomorphism of algebras

holds

H u u kP J H Z d

mn P

COHOMOLOGY OF MOMENTANGLE COMPLEXES

where

bideg v bideg u

i i

du w dv

i i i

mn n

Proof Since H T u u and H M kP J we

have

E H kP J u u d

By Lemma

H M k H kP J u u d Tor

kw w

1 mn

Combining the ab ove two identities with Lemma we get E H Z which

concludes the pro of

Our next aim is to calculate the cohomology of the quotient Z H for arbitrary

freely acting subgroup H First we write H in the form and cho ose an

m r mmatrix T t of rank m r satisfying T S This is done in

the same way as in the pro of of Prop osition In particular if r m n then

T is the characteristic matrix for the quasitoric manifold Z H

Theorem The fol lowing isomorphism of algebras holds

H Z H Tor kP k

kt t

1 mr

where the kt t module structure on kP kv v I is given by

mr m P

the map

k t t k v v

mr m

t t v t v

i i im m

Remark Theorem reduces to Theorem in the case r and to

Example in the case r m n

r m

Proof of Theorem The inclusion T H T denes the map

r m

h BT BT of the classifying spaces Let us consider the commutative square

E B P

y y

r m

BT BT

where the left vertical arrow is the pullback along h The space E is homotopy

equivalent to the quotient Z H Hence the Eilenb ergMo ore sp ectral sequence

of the ab ove square converges to the cohomology of Z H Its E term is

E Tor kP kw w

kv v

1 m

where the kv v mo dule structure in kw w is dened by the ma

m r

trix S ie by the map v s w s w In the same way as in the pro of of

i i ir r

Theorem we show that the sp ectral sequence collapses at the E term and the

following isomorphism of algebras holds

H Z H Tor kP kw w

P r

kv v

1 m

BIGRADED POINCARE DUALITY AND DEHNSOMMERVILLE EQUATIONS

Now put kv v kt t A kw w and C kP

m mr r

in Theorem Since is a free mo dule and kw w a

sp ectral sequence fE d g arises Its E term is

s s

kP k E Tor kw w Tor

kw w kt t

1 r 1 mr

and it converges to Tor kP kw w Obviously kw w is

r r

kv v

1 m

a free kw w mo dule so we have

e e

E for p E Tor kP k

kt t

1 mr

Thus the sp ectral sequence collapses at the E term and the following isomorphism

of algebras holds

kP k kP kw w Tor Tor

kt t kv v

1 mr 1 m

which together with concludes the pro of

Corollary H Z H H u u kP d where du

P mr i

t v t v dv bideg v bideg u

i im m i i i

Example Let H S is the diagonal subgroup Then the matrix S is

a column of m units By Theorem

H Z S Tor kP k

P d

kt t

1 m1

where the kt t mo dule structure in kP kv v I is dened

m m

t v v i m

i i m

Supp ose that the S bundle Z Z S is classied by a map c Z S

P P d P d

BT C P Since H C P kw the element c w H Z S is dened

P d

n q

Lemma P is q neighborly if and only if c w

Proof The map c takes the cohomology ring H BT kw to the sub

algebra

kP k H Z H kP k Tor

kt t

1 m1

This subalgebra is isomorphic to the quotient kP v v Now the

assertion follows from the fact that a p olytop e P is q neighb orly if and only if the

ideal I do es not contain monomials of degree q

Bigraded Poincare duality and DehnSommerville equations

Here we assume that K is a triangulated manifold In this case the corre

sp onding momentangle complex Z is not a manifold in general however its singu

larities can b e easily treated Indeed the cubical complex ccK Construction

is homeomorphic to j coneK j and the vertex of the cone is p

ccK I Let U p ccK b e a small neighb orho o d of p in ccK The clo

sure of U p is also homeomorphic to j coneK j It follows from the denition of

Z see that U Z U p Z is a small invariant neighb orho o d

K K

T in Z For small the closure of U Z is home of the torus Z p

omorphic to j coneK j T Removing U Z from Z we obtain a manifold

with b oundary which we denote W Thus we have

W Z n U Z W jK j T

K K K

m m

Note that since U Z is a T stable subset the torus T acts on W

COHOMOLOGY OF MOMENTANGLE COMPLEXES

Theorem The manifold with boundary W is equivariantly homotopy

equivalent to the momentangle complex W see Also there is a canonical

relative homeomorphism of pairs W W Z Z

K K K

Proof To prove the rst assertion we construct homotopy equivalence ccK n

U p cubK as it is shown on Figure This map is covered by an equivariant

homotopy equivalence W Z n U Z W The second assertion follows

K K K

easily from the denition of W

u u

ccK n U p

X X

cubK J

X X

J C

u CW

u u

Figure Homotopy equivalence ccK n U p cubK

By Lemma the momentangle complex W D has a cellular struc

ture with dierent cell typ es D I T i m see Figure The

i i i i i

homology of W and therefore of W can b e calculated from the corresp onding

K K

cellular chain complex which we denote C W Although W has more

K c K

typ es of cells than Z instead of its cellular chain complex C W

K K c

also has a natural bi grading Namely the following statement holds compare

with

Lemma Put

bideg D bideg T bideg I

i i i

bideg bideg i m

i i

This turns the cel lular chain complex C W into a bigraded dierential mod

K c

ule with dierential adding to bidegree The original grading of C W

c K

by the dimension of cel ls corresponds to the total degree ie the dimension of a

cel l equals the sum of its two degrees

Proof The only thing we need to check is that the dierential adds

to bidegree This follows from and the following formulae

D T I T

c i i c i i i c i c i c i

Unlike the bigraded structure in C Z elements of C W may have positive

K K

rst degree due to the p ositive rst degree of I The dierential do es not

i c

change the second degree as in the case of Z which allows us to split the

bigraded complex C W into the sum of complexes C W p m

K p K

BIGRADED POINCARE DUALITY AND DEHNSOMMERVILLE EQUATIONS

In the same way as we did this for Z and Z Z we dene

K K

b W dim H C W m 6 q 6 m 6 p 6 m

q p K q p K c

m m

X X

q q

W dim C W b W

p K q p K q p K

q m q m

W t W t

K p K

note that q ab ove may b e negative

The following theorem gives a formula for the generating p olynomial W t

and is analogous to theorems and

Theorem For any simplicial complex K with m vertices it holds that

mn n m

W t t h h t h t K t

K n

mn n n m

t h h t h t h t

n n

where K f f f h is the Euler charac

n n

teristic of K

Proof By the denition of W see the vector R fD I T g

see section represents a cell of W if and only if the following two conditions

are satised

a The set R R R is a simplex of K

D I

b jR j >

Let c W denote the numb er of cells R W with jR j i jR j j

ij lpq K K D I

jR j l jR j p jR j q i j l p q m It follows that

ij l j l mij l

c W f

ij lpq K ij l

i l p

where f f is the f vector of K we also assume f and f for

n k

k or k n By

bideg R jR j jR j jR j jR j j p i p

I T D T

Now we calculate W using and

r K

ij l j l mij l

j p

W f

r K ij l

i l p

ijlp

iprl>

Substituting s i j l ab ove we obtain

s sr p ms

sr l

W f

r K s

r p l p

lsp

X X

s ms sr p

sr l

r p p l

Since

s r p

sr p

s 6 r p

COHOMOLOGY OF MOMENTANGLE COMPLEXES

we get

ms s

f W

s r K

p r p

p sr

X X

ms ms s

r s r s

f f

s s

r s p r p

s sp

The second sum in the ab ove formula is exactly Z see To calculate

r K

the rst sum we observe that

s ms m

r p p r

This follows from calculating the co ecient of in the two sides of the identity

s ms m

Hence

m m

r s r

W f Z K Z

r K s r K r K

r r

since f K Finally using we calculate

m m m

X X X

r r r r

K t Z t W t W t

r K K r K

r r r

m mn n

K t t h h t h t

Supp ose that K is an orientable triangulated manifold It is easy to see that

then W is also orientable Hence there are relative Poincare duality isomorphisms

mnk

H W H W W k m

k K K K

Theorem DehnSommerville equations for triangulated manifolds

The fol lowing relations hold for the hvector h h h of any triangulated

manifold K

n n i

K S i n h h

ni i

n n

where S is the Euler characteristic of an n sphere

Proof Supp ose rst that K is orientable By Theorem H W

k K

mnk mnk

H W and H W W H Z Z Moreover it can b e

k K K c K K

seen in the same way as in Corollary that relative Poincare duality isomor

phisms regard the bigraded structures in the cohomology of W and

Z Z Hence

b W b Z Z

q p K K

mnq mp

W Z Z

p K mp K

mn m

W t t Z Z

K K t

BIGRADED POINCARE DUALITY AND DEHNSOMMERVILLE EQUATIONS

Using we calculate

mn m

t Z Z

mn m mn n

t t h h t h t

mn m m

t t

mn n n n m

t h t h t h t

Substituting the formula for W t from Theorem and the ab ove expression

into we obtain

mn n m

t h h t h t K t

mn n n n m

t h t h t h t

Calculating the co ecient of t in b oth sides after dividing the ab ove identity by

mn i n n

t we get h h K S as required

ni i

Now supp ose that K is nonorientable Then there exist an orientable triangu

lated manifold L of the same dimension and a sheet covering L K Then we

obviously have f L f K i n It follows from that

i i

n n

X X

ni n ni n

h Lt t h K t t

i i

Hence

i n h L h K

i i

i n

Since L is orientable we have h L h L L S There

ni i

fore

n n n

n i i ni

L S h K h K

ni i

i i ni

Since L K we get

n n i

K S h K h K

ni i

n i

K S

as required

If jK j S or n is o dd then Corollary gives the classical equations

h h

ni i

Corollary Suppose K is a triangulated manifold with the hvector

h h Then

h h h i n

ni i n

n n n n

Proof Since K h and S we have

n n n

K S h h

n n

the co ecient can b e dropp ed since for o dd n the left hand side is

zero

Corollary For any n dimensional triangulated manifold the num

bers h h i n are homotopy invariants In particular they do not

ni i

depend on a triangulation

COHOMOLOGY OF MOMENTANGLE COMPLEXES

In the case of PLmanifolds the top ological invariance of numb ers h h

ni i

was observed by Pachner in

b f h a f h

Figure Symmetric and minimal triangulation of T

Example Triangulations of manifolds Consider triangulations of the

n n

torus T We have n T From K h we deduce

h Corollary gives

h h h h

For instance the triangulation on Figure a has f vertices f edges

and f triangles Note that this triangulation is the canonical triangulation

of as describ ed in Construction The corresp onding hvector is

On the other hand it is well known that a triangulation of T with only

vertices can b e achieved see Figure b Note that this triangulation is neigh

b orly ie its skeleton is a complete graph on vertices It turns out that no

triangulation of T with smaller numb er of vertices exists

Supp ose now K is a dimensional triangulated manifold with m vertices Let

K b e its Euler characteristic Using Corollary we may express the

f vector of K via and m namely

f K m m m

Since the numb er of edges in a triangulation do es not exceed the numb er of pairs

of vertices we get the inequality

m 6 mm

from which a lower b ound for the numb er of vertices in a triangulation of K can

b e deduced For instance in the case of torus T we have and gives

m > Note that a minimal triangulation of K is neighb orly has a complete

graph as its skeleton only if turns to equality We have seen that this

is the case for T m Other examples are the sphere S

m and the real pro jective plane RP m A neighb orly triangu

lation of RP is shown in Figure However for most values of there is no m

which makes an equality For example minimal triangulations of orientable

surfaces of genus to are not neighb orly A genus surface having

and m has neighb ourly triangulations which are automatically minimal

These triangulations are imp ortant in the problem of p olyhedral emb eddability of

BIGRADED POINCARE DUALITY AND DEHNSOMMERVILLE EQUATIONS

A A

Figure Neighb orly triangulation of RP with f

orientable triangulated surfaces in R p olyhedral here means with at triangles

and no selfintersections It was shown in that there are in total dier

ent neighb orly triangulations of a genus surface with vertices Later using

an algorithm for generating oriented matroids Bokowski and Guedes de Oliveira

proved in that one of these triangulations cannot b e emb edded into R with

at triangles Furthermore they proved that one triangle can b e removed from the

triangulation while retaining nonemb eddability so an arbitrary numb er of handles

can b e attached at this triangle to get a nonemb eddable triangulated surface of any

genus > A numb er of results on minimal triangulations were obtained by Lutz

in using his computer program BISTELLAR which we already mentioned in

section

COHOMOLOGY OF MOMENTANGLE COMPLEXES

CHAPTER

Cohomology rings of subspace arrangement

complements

General arrangements and their complements

Definition An arrangement is a nite set A fL L g of ane

subspaces in some ane space either real or complex An arrangement A is called

a subspace arrangement or central arrangement if all its subspaces are linear ie

contain Given an arrangement A fL L g in C dene its support or

union jAj as

jAj L C

and its complement U A as

U A C n jAj

and similarly for arrangements in R

Let A fL L g b e an arrangement The intersections

v L L

i i

with resp ect to the inclusion called the intersection poset of form a p oset L

the arrangement The p oset L is assumed to have a unique maximal element T

corresp onding to the ambient space of the arrangement The rank function d on L

is dened by dv dim v The complex ordL see Example is called the

order complex of arrangement A Dene intervals

L fx L v x w g L fx L x v g

v w

Arrangements and their complements play a pivotal role in many construc

tions of combinatorics algebraic and symplectic geometry etc they also arise as

conguration spaces for dierent classical mechanical systems In the study of ar

rangements it is very imp ortant to get a suciently detailed description of the

top ology of complements U A this includes numb er of connected comp onents

homotopy typ e homology groups cohomology ring etc A host of elegant results

in this direction app eared during the last three decades however the whole picture

is far from b eing complete The theory ascends to work of Arnold in which

the classifying space for the colored braid group is describ ed as the complement

of the arrangement of all diagonal hyp erplanes fz z g 6 i j 6 n in C

i j

The latter complement can b e thought as the conguration space of n ordered

p oints in C Its cohomology ring was also calculated in This result was gen

eralized by Brieskorn and motivated the further development of the theory of

complex hyperplane arrangements ie arrangements of co dimensionone complex

ane subspaces One of the main results here is the following

COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

Theorem Let A fL L g be an arrangement of complex

hyperplanes in C where the hyperplane L is the zero set of linear function l

j j

j r Then the integer cohomology algebra of the complement C n jAj is

isomorphic to the algebra generated by closed dierential forms

i l

Relations b etween the forms j r were explicitly de

i l

scrib ed by Orlik and Solomon We give their result in the central case ie

when all the hyp erplanes are vector subspaces Then there is one relation

j j j

1 p

for any minimal subset fL L g of hyp erplanes of A such that co dim L

j j j

1 p 1

L p such subsets are called circuits of L

Example Let A b e the arrangement of diagonal hyp erplanes fz z g

j k

z dz

6 j k 6 n in C Then we have the forms satisfying the

j k

i z z

identities

ij j k j k k i k i ij

known as the Arnold relations

The theory of complex hyp erplane arrangements is probably the most well un

dersto o d part of the whole study Several surveys and monographs are available we

mention just and where further references can b e found Rela

tionships b etween real hyp erplane arrangements p olytop es and oriented matroids

are discussed in Lecture Another interesting related class of arrangements

is known as arrangements in R A arrangement is an arrangement of real

subspaces of co dimension with evendimension intersections In particular any

complex hyp erplane arrangement is a arrangement The relationships b etween

arrangements and complex hyp erplane arrangements are studied in

In the case of general arrangement A the celebrated GoreskyMacPherson

theorem Part I I I expresses the cohomology groups H U A without ring

structure as a sum of homology groups of sub complexes of a certain simplicial

complex

Theorem Goresky and MacPherson Part I I I The cohomology of

a subspace arrangement complement U A in R is given by

ndv i

ordL ordL Z H H U A Z

v i

v T

v P

U A Z H ordL ordL Z H

ndv i v T

v P

see Denition where we assume that H H Z

The original pro of of this theorem used the stratied Morse theory develop ed

Remark Observing that ordL is the cone over ordL we may rewrite

v T

the formula from Theorem as

ndv i

e e

H U A Z H ordL Z

v T

v P

and similarly for the cohomology

COORDINATE SUBSPACE ARRANGEMENTS

Remark The homology groups of a complex arrangement in C can b e cal

culated by regarding it as a real arrangement in R

A comprehensive survey of general arrangements is given in Mono

graph gives an alternative approach to homology and homotopy computa

tions via the Anderson spectral sequence A metho d of describing the homotopy

typ es of subspace arrangements using diagrams of spaces over poset categories

was prop osed in Using this metho d a new elementary pro of of Goresky

MacPherson Theorem was found there The approach of was develop ed

later in by incorp orating homotopy colimits techniques

The cohomology rings of arrangement complements are much more subtle In

general the integer cohomology ring of U A is not determined by the intersection

p oset L this is false even for arrangements as shown in An approach to

calculating the cohomology algebra of the complement U A based on the results

of De Concini and Pro cesi was prop osed by Yuzvinsky in Recently

the combinatorial description of the pro duct of any two cohomology classes in a

complex subspace arrangement complement U A conjectured by Yuzvinsky has

b een obtained indep endently in and This description is given in terms

of the intersection p oset LA the dimension function and additional orientation

data

Co ordinate subspace arrangements and the cohomology of Z

An arrangement A fL L g is called coordinate if every L i r

r i

is a co ordinate subspace In this section we apply the results of chapter to

cohomology algebras of complex co ordinate subspace arrangement complements

The case of real co ordinate arrangements is also discussed at the end of this section

A co ordinate subspace of C can b e written as

L fz z C z z g

m i i

where fi i g is a subset of m Obviously dim L m j j

Construction For each simplicial complex K on the set m dene the

complex co ordinate subspace arrangement CAK by

CAK fL K g

Denote the complement of CAK by U K that is

U K C n L

Note that if K K is a sub complex then U K U K

Proposition The assignment K U K denes a onetoone order

preserving correspondence between the set of simplicial complexes on m and the

m m

set of coordinate subspace arrangement complements in C or R

Proof Supp ose CA is a co ordinate subspace arrangement in C Dene

K CA f m L jC Ajg

Obviously K CA is a simplicial complex By the denition K CA dep ends only

on jC Aj or U CA and U K CA U CA whence the prop osition follows

COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

If CA contains a hyp erplane say fz g then its complement U CA is

factored as U CA C where CA is a co ordinate subspace arrangement in

the hyp erplane fz g and C C n fg Thus for any co ordinate subspace

arrangement CA the complement U CA decomp oses as

U CA U CA C

were CA is a co ordinate arrangement in C that do es not contain hyp erplanes

On the other hand shows that CA contains the hyp erplane fz g if and

only if fig is not a vertex of K CA It follows that U K is the complement of a

co ordinate arrangement without hyp erplanes if and only if the vertex set of K is the

whole m Keeping in mind these remarks we restrict our attention to co ordinate

subspace arrangements without hyp erplanes and simplicial complexes on the vertex

set m

Remark In the notations of Construction we have U K K C C

m m

Example If K then U K C

m m

If K b oundary of simplex then U K C n fg

If K is a disjoint union of m vertices then U K is the complement in C

of the set of all co dimensiontwo co ordinate subspaces z z 6 i j 6 m

i j

m m

The diagonal action of algebraic torus C on C descends to U K In

m m

particular there is the standard action of T on U K The quotient U K T

m m m

is regarded as a subset of C where R can b e identied with U K R

and Z U K see Construction Lemma ccK U K R

and

Proof Take y y y ccK Let fi i g b e the set of

m k

zero co ordinates of y ie the maximal subset of m such that y L R

Then it follows from the denition of ccK see that is a simplex of K

Hence L CAK and y U K which implies the rst statement The second

assertion follows from the fact that ccK is the quotient of Z

Theorem There is an equivariant deformation retraction U K Z

ccK Proof First we construct a deformation retraction r U K R

This is done inductively We start from the b oundary complex of an m simplex

and remove simplices of p ositive dimensions until we obtain K On each step we

construct a deformation retraction and the comp osite map will b e the required

retraction r

If K is the b oundary complex of an m simplex then U K

m m

n fg In this case the retraction r is shown on Figure Now supp ose R R

that K is obtained from K by removing one k dimensional simplex

fj j g that is K K By the inductive hyp othesis we may assume that

m m

there is a deformation retraction r U K R ccK Let a R b e the

p oint with co ordinates y y and y for i Since is not

j j i

a simplex of K we have a U K R At the same time a C see

Hence we can apply the retraction shown on Figure on the face C I

with center at a Denote this retraction by r Then r r r is the required

deformation retraction

The deformation retraction r U K R ccK is covered by an equivariant

deformation retraction U K Z which concludes the pro of K

COORDINATE SUBSPACE ARRANGEMENTS

u u

u e

m m

Figure The retraction r U K R ccK for K

In the case K K ie K is a p olytopal simplicial sphere corresp onding to

a simple p olytop e P the deformation retraction U K Z from Theorem

P P

can b e realized as the orbit map for an action of a contractible group We denote

U P U K Set

m m m

fy y R y i mg R R

m i

m m m

is a group with resp ect to the multiplication and it acts on R C and Then R

n m

U P by co ordinatewise multiplications There is the isomorphism exp R

m m

b etween the additive and the multiplicative groups taking y y R R

y y m

1 m

to e e R

Let us consider the m m nmatrix W intro duced in Construction for

every simple p olytop e

Proposition For any vertex v F F of P the maximal minor

i i

1 n

W of W obtained by deleting n rows i i is nondegenerate det W

i i i i

1 n 1 n

Proof If det W then the vectors l l see are linearly

i i

1 n

i i

1 n

dep endent which is imp ossible

The matrix W denes the subgroup

w w w w m

11 1 1mn mn m1 1 mmn mn

e e R R

where is running through R Obviously R R

mn W

Theorem Theorem and x The subgroup R acts freely

n m n n

on U P C The composition Z U P U P R of the embedding

P W

i Lemma and the orbit map is an equivariant dieomorphism with respect

to the corresponding T actions

Supp ose now that P is a lattice simple p olytop e and let M b e the corre

sp onding toric variety Construction Along with the real subgroup R

R dene its complex analogue

w w w w m

11 1 1mn mn m1 1 mmn mn

C e e C

mn mn

where is running through C Obviously C C It is

mn W

shown in see also that C acts freely on U P and the toric variety W

COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

M can b e identied with the orbit space or geometric quotient U P C

P W

Thus we have the following commutative diagram

R R

U P Z

C C

y y

W T

M M

P P

Remark It can b e shown Theorem that any toric variety M cor

resp onding to a fan R with m onedimensional cones can b e identied with

the universal categorical quotient U CA G where U CA a certain co ordinate

arrangement complement determined by the fan and G C The cate

gorical quotient b ecomes the geometric quotient if and only if the fan is simplicial

In this case U CA U K

On the other hand if the pro jective toric variety M is nonsingular then M

P P

is a symplectic manifold of dimension n and the action of T on it is Hamiltonian

see eg or x In this case the diagram displays M as the result of a

symplectic reduction Namely let H T b e the maximal compact subgroup

m mn

in C and C R the moment map for the Hamiltonian action of H

W W

m mn

on C Then for any regular value a R of the map there is the following

dieomorphism

aH U P C M

W W P

details can b e found in In this situation a is exactly our manifold Z

This gives us another interpretation of the manifold Z as the level surface for the

moment map in the case when P can b e realized as the quotient of a nonsingular

pro jective toric variety

n n n

Example Let P the nsimplex Then m n U P

C n fg Moreover R R C C and H S are the diagonal

W W W

n m

C and T resp ectively see Example Hence subgroups in R

n n n n

Z S C n fg R M C n fg C C P

P P

m m

The moment map C R takes z z C to jz j jz j

m m

and for a we have a S Z

Now we have the following result for the cohomology of subspace arrangement

complements

Theorem The fol lowing isomorphism of graded algebras holds

H Tor U K kK k

kv v

1 m

H u u kK d

Proof This follows from Theorems and

Theorem provides an eective way to calculate the cohomology algebra

of the complement of any complex co ordinate subspace arrangement The Koszul

complex was also used by De Concini and Pro cesi and Yuzvinsky for

constructing rational mo dels of the cohomology algebra of an arrangement com

plement As we see in the case of co ordinate subspace arrangements calculations

COORDINATE SUBSPACE ARRANGEMENTS

b ecome shorter and more eective as so on as the StanleyReisner ring is brought

into the picture

Problem Calculate the integer cohomology algebra of a coordinate sub

space arrangement complement and compare it with the corresponding Toralgebra

Tor ZK Z

Zv v

1 m

Example Let K b e a disjoint union of m vertices Then U K is the

complement to the set of all co dimensiontwo co ordinate subspaces z z

i j

6 i j 6 m in C see Example The face ring is kK kv v I

m K

where I is generated by the monomials v v i j An easy calculation using

K i j

Corollary shows that the subspace of co cycles in kK u u has

the basis consisting of monomials v u u u with k > and i i for

i i i i p q

1 2 3

p q Since deg v u u u k the space of k dimensional co cycles

i i i i

1 2 3

m m

The space of k dimensional cob oundaries is has dimension m

k k

dimensional it is spanned by the cob oundaries of the form du u Hence

i i

dim H U K H U K H U K

m m m

dim H U K m k 6 k 6 m

k k k

and the multiplication in the cohomology is trivial

In particular for m we have threedimensional cohomology classes v u

i j

i j sub ject to relations v u v u and fourdimensional cohomology

i j j i

classes v u u v u u v u u sub ject to one relation

v u u v u u v u u

Hence dim H U K dim H U K and the multiplication is trivial

It can b e shown that U K in this case has a homotopy typ e of a wedge of spheres

U K S S S S S

Example Let K b e the b oundary of an mgon m Then

U K C n fz z g

i j

ij mo d m

By Theorem the cohomology ring of H U K k is isomorphic to the ring

describ ed in Example note that the multiplication is nontrivial here

As it is shown in in the case of arrangements of real co ordinate subspaces

only additive analogue of our Theorem holds Namely let us consider the

p olynomial ring kx x with deg x i m Then the graded

m i

structure in the face ring kK changes accordingly The Betti numb ers of the

real co ordinate subspace arrangement U K can b e calculated by means of the

following result

Theorem Theorem The fol lowing isomorphism holds

p ij

H Tor U K kK k H u u kK d

R m

kx x

1 m

ij p

where bideg u bideg v du x dx

i i i i i

As it was observed in there is no multiplicative isomorphism analogous to

Theorem in the case of real arrangements that is the algebras H U K and

Tor kK k are not isomorphic in general The pap er also contains

kx x

1 m

COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

the formulation of the rst multiplicative isomorphism of our Theorem for

complex co ordinate subspace arrangements see Theorem with reference

to a pap er by Babson and Chan unpublished

Up to this p oint we have used the description of co ordinate subspaces by means

of equations see On the other hand a co ordinate subspace can b e dened

as the linear span of a subset of the standard basis fe e g This leads to

the dual approach to co ordinate subspace arrangements which corresp onds to the

passage from simplicial complex K to the dual complex K Example Namely

we have

CAK spanfe e g fi i g K

i i k

see Construction We may observe further that in the co ordinate subspace

arrangement case the intersection p oset L is the inclusion p oset of simplices

b b

of K with added maximal element or equivalently the inclusion p oset of cone K

v where v is regarded as Hence ordL is the barycentric sub division of link

v T

a simplex of K Thus we may rewrite the GoreskyMacPherson formula in

the complex subspace arrangement case as

mj ji

e e

H link H U K

note that d j j and the dimensions are doubled since we are in the complex

arrangement case

The ab ove observations were used in to describ e the pro duct of two co

homology classes of a co ordinate subspace arrangement complement either real or

complex in terms of the combinatorics of links of simplices in K see Theo

rem

On the other hand the isomorphism of algebras established in Theorem al

lows us to connect two seemingly unrelated results namely the GoreskyMacPher

son theorem for the cohomology of an arrangement complement and the Ho chster

theorem from the commutative algebra

Proposition After identication of the cohomology H U K with the

Toralgebra Tor kK k established by Theorem the Hochster The

kv v

1 m

orem becomes equivalent to the GoreskyMacPherson Theorem in the case

of coordinate subspace arrangements

ij ij

Proof Using Theorem to identify kK with dim H U K

we get the following formula from Ho chsters Theorem

H U K H K

pj j

Nonempty simplices K do not contribute to the ab ove sum since the corre

sp onding full sub complexes K are contractible Since H k the empty

subset of m only contributes k to H U K Hence we may rewrite the ab ove

formula as

e e

H U K H K

pj j

Using the Alexander duality Prop osition we calculate

mpj jjbj mjbjp

e e e

H K H link b H link b

b b

pj j

K K

DIAGONAL SUBSPACE ARRANGEMENTS

where b m n is a simplex of K Now we observe that is equivalent

Diagonal subspace arrangements and the cohomology of Z

Another interesting particular class of subspace arrangements is diagonal ar

rangements A classical example of a diagonal subspace arrangement is given by the

arrangement of all diagonal hyp erplanes fz z g in C mentioned in section

i j

see Example Some further particular examples of diagonal arrangements the

socalled k equal arrangements were considered eg in while the cohomology

of general diagonal arrangement complements was studied in In this section

we establish certain relationships b etween this cohomology and the cohomology of

the lo op spaces B Z see section and Z

T K K

Definition For each subset fi i g m dene the diagonal

subspace D in R by

D fy y R y y g

m i i

Diagonal subspaces in C are dened similarly An arrangement A fL L g

is called diagonal if all L i r are diagonal subspaces

Construction Given a simplicial complex K on the vertex set m

intro duce the diagonal subspace arrangement D AK as the set of subspaces D

such that is not a simplex of K

DAK fD K g

Denote the complement of the arrangement DA K by M K

The following statement is proved in the similar way as the corresp onding

statement Prop osition for co ordinate subspace arrangements

Proposition The assignment K M K denes a onetoone order

preserving correspondence between the set of simplicial complexes on the vertex

set m and the set of diagonal subspace arrangement complements in R

Here we still assume that k is a eld The multigraded or N graded struc

ture in the ring kv v Construction denes an N grading in the

v acquires the multidegree StanleyReisner ring kK The monomial v

i i Let us consider the mo dules Tor k k They can b e calcu

lated by means of the minimal free resolution Example of k regarded as a

kK mo dule The minimal resolution also carries a natural N grading and we

denote the subgroup of elements of multidegree i i in Tor k k by

Tor k k

kK i i

1 m

Theorem Theorem The fol lowing isomorphism holds for the

cohomology groups of a real diagonal subspace arrangement complement M K

H Tor M K k k k

Remark Instead of simplicial complexes K on the vertex set m the authors

of considered squarefree monomial ideals I kv v Prop osition

shows that the two approaches are equivalent

COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

Theorem The fol lowing additive isomorphism holds

H B Z k Tor k k

T K

Proof Let us consider the Eilenb ergMo ore sp ectral sequence of the Serre

bration P DJ K with bre DJ K where DJ K is the DavisJanuszkiewicz

space Denition and P is the path space over DJ K By Corollary

H P k Tor k k E Tor

kK K H DJ

and the sp ectral sequence converges to Tor C P k H DJ K

C DJ K

Since P is contractible there is a co chain equivalence C P k We have

C DJ K kK Therefore

Tor C P k Tor k k

C DJ K kK

which together with shows that the sp ectral sequence collapses at the E

term Hence H DJ K Tor k k Finally Theorem shows that

H DJ K H B Z which concludes the pro of

T K

Proposition The fol lowing isomorphism of algebras holds

H B Z H Z u u

T K K m

Proof Consider the bundle B Z BT with bre Z It is easy to see

T K K

that the corresp onding lo op bundle B Z T with bre Z is trivial note

T K K

m m m

that BT T To nish the pro of it remains to mention that H T

u u

Theorems and give an application of the theory of momentangle com

plexes to calculating the cohomology ring of a co ordinate subspace arrangement

complement Likewise Theorems and Prop osition establish a con

nection b etween the cohomology of a diagonal subspace arrangement complement

and the cohomology of the lo op space over the momentangle complex Z How

ever the latter relationships are more subtle than those in the case of co ordinate

subspace arrangements For instance we do not have an analogue of the multi

plicative isomorphism from Theorem It would b e very interesting to get any

statement of such kind or discover other new applications of the theory of moment

angle complexes to diagonal or mayb e even general subspace arrangements

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Index

complex Ane equivalence

Alexander duality

oriented

simplicial

CohenMacaulay

Ample divisor

complex

Arithmetic genus

ring algebra

Arnold relations

Colimit

Arrangement

coloring

diagonal

Combinatorial

co ordinate

equivalence

hyp erplane

neighb orho o d

k equal

Complement of an arrangement

subspace central

Cone

Artin group

convex p olyhedral

nonsingular

Bistellar

rational

equivalence

simplicial

moves ips op erations

strongly convex

Barycenter

Connected sum Barycentric sub division

Bigraded

of simple p olytop es

Betti numb ers

of simplicial complexes

dieren tial mo dule

Convex p olyhedron

dierential algebra

Core

Blowup blowdown

Coxeter group

Borel construction

Coxeter complex

Boundary

Crossp olytop e

Bounded ag manifold

Cub e

standard

Chain

top ological

Characteristic map

Cubical

directed

complex

Characteristic pair

abstract

directed

combinatorialgeometrical

CharneyDavis conjecture

emb eddable into lattice

Chow ring

sub division

Chromatic numb er

of simple p olytop e Cob ordisms

INDEX

DavisJanuszkiewicz space ring algebra

DehnSommerville relations Graph pro duct

for triangulated manifolds

hvector

Dehn twist

of algebra

Depth

of p olytop e

Dimension

of simplicial complex

homological

Hard Lefschetz theorem

Krull

Hauptvermutung der Top ologie

of cubical complex

Hilb ert series

of p olytop e

Hinge mechanisms

of simplicial complex

Hirzebruch genus

Dolb eault complex

Hirzebruch surface

Edge

Hopf conjecture

Edge vector

Homotopy colimit

Elementary shellings

hsop homogeneous system of parameters

Eulerian complex

Face

Ideal

missing

monomial

of convex p olyhedral cone

StanleyReisner

of cubical complex

Index of a vertex

of p olytop e

Intersection cohomology

of p olyhedron

Intersection hvector

prop er

Intersection p oset of an arrangement

Face p oset

Face ring

Join

of simple p olytop e

genus L

of simplicial complex

Link

Facet

lsop linear system of parameters

Facet vector

Lower Bound Conjecture LBC

Facial submanifold

generalized GLBC

Factorization conjecture strong weak

Fan

Manifold

complete

PL combinatorial

nonsingular

quasitoric

normal

nontoric

p olytopal

stably complex

simplicial

toric

strongly p olytopal

unitary

weakly p olytopal

triangulated simplicial

Four Color Theorem

with corners

Flag complex

Milnor hyp ersurfaces

Flagication

Milnor ltration

Flip

Minimal

f vector

generator set basis

of cubical complex

map

of p olytop e

resolution

of simplicial complex

Mirroring construction

Momentangle

conjecture g

manifold

g theorem

complex

g vector

Moment curve of p olytop e

Moment map

of simplicial complex

Morse theory Geometrical realization

stratied

Ghost vertex

Multifan Gorenstein Gorenstein

M vector complex

INDEX

abstract No ether normalization lemma

geometrical

Omniorientation

standard

Orbifold

regular

Order complex

Simplicial

of arrangement

complex

Oriented matroid

abstract

dual

Piecewise linear PL

geometrical

homeomorphism

k neighb orly

manifold

pure

with b oundary

underlying of a fan

map

fan

sphere

isomorphism

Poincare series

manifold

Polar set

map

Polyhedron

nondegenerate

convex

sphere

Polytop e

sub division

combinatorial

of cub e

convex

stellar

cyclic

Skeletal rigidity

generic

Small cover

geometrical

Sp ectral sequence

k neighb orly

Anderson

lattice

Eilenb ergMo ore

neighb orly

LeraySerre

nonrational

Sphere

p olar dual

Barnette

rational

Bruc kner

simple

homology

simplicial

nonPL

stacked

nonp olytopal

Polytop e algebra

Poset

Poincare

Eulerian

p olytopal

Poset category

simplicial

Positive cone

stacked

Pro duct

Stably complex

of simple p olytop es

manifold

of simplicial complexes

structure

Pseudomanifold

canonical

Pullback from the linear mo del

StanleyReisner ring

of simple p olytop e

Quadratic algebra

of simplicial complex

Rank function

Star

Regular sequence

Sub complex

Resolution

full

free

cubical

Koszul

simplicial

minimal

Supp ort of an arrangement

pro jective

Supp orting hyp erplane

Ray

Surgery

equivariant

Schlegel diagram

Susp ension

Serre problem

Symplectic reduction

Sign of a vertex

Signature

Tangent bundle

T Simplex manifold

INDEX

To dd genus

Toralgebra

Toric variety

Torus

algebraic

Torus action

Hamiltonian

lo cally standard

standard

Triangulation Conjecture

Union of an arrangement

Upp er Bound Conjecture UBC

for p olytop es

for simplicial spheres

Vertex

Vertex set

Volume p olynomial

arrangement

genus

equivariant

translation