Homotopy Colimits Comparison Lemmas for

Combinatorial Applications

Volkmar Welker

Fachb ereich Mathematik

Universitat GHEssen

Essen Germany

welkerexpmathuniessende

Gun ter M Ziegler

Department of Mathematics

Technical University of Berlin

Berlin Germany

zieglermathtuberlinde

Rade T Zivaljevic

Mathematics Institute

Knez Mihailova PF

Belgrade Yugoslavia

ezivaljeubbgetfbgacyu

Octob er

Abstract

We provide a to olkit of basic lemmas for the comparison of homotopy typ es

of homotopy colimits of diagrams of spaces over small categories We show how this

to olkit can b e used on quite dierent elds of applications We demonstrate this

with resp ect to

Bjorners Generalized Homotopy Complementation Formula

the top ology of toric varieties

the study of homotopy typ es of arrangements of subspaces

the analysis of homotopy typ es of subgroup complexes

1

Supp orted by a Habilitationsstip endium of the DFG

2

Supp orted by a GerhardHessForderpreis of the DFG which also provided supp ort for the authors

joint work at the KonradZuseZentrum fur Informationstechnik Berlin ZIB in

3

Supp orted by the Mathematical Institute SANU Belgrade grant no D

Contents

Intro duction

Fundamental concepts and constructions

Basic denitions and motivating examples of diagrams

Limit space constructions

Simplicial homotopy lemma and the gluing lemma

Comparison results for diagrams of spaces

Pro jection wedge and the quasibration lemma

From lo cal to global homotopy equivalences

Homotopies arising from natural transformations

Diagrams over p osets

Homotopy colimits revisited

Comparison results for diagrams over p osets

Applications

Combinatorial ob jects as homotopy colimits

Bjorners generalized homotopy complementation formula

Toric varieties

Subspace arrangements

Subgroup complexes

Acknowledgments

References

Intro duction

The aim of this pap er is to advertise homotopy colimit considerations for top ological inves

tigations in combinatorics For this we provide a to olkit and demonstrate its usefulness

by a numb er of applications

A diagram of spaces is a functor from a small category to the category of top ological

spaces In various top ological geometric algebraic and combinatorial situations one

has to deal with structures that can protably b e interpreted as colimits or homotopy

colimits of diagrams over small categories sp ecially over nite p osets In fact if a space

is written as a nite union of simpler pieces then it is the colimit of a corresp onding

diagram of spaces While colimits do not have go o d functorial prop erties in homotopy

theory they can usually b e replaced by homotopy colimits Pupp e may have b een

the rst to exploit this Homotopy colimits have much b etter functorial prop erties

Thus there is a wide variety of techniques to manipulate diagrams of spaces in such a way

that the homotopy colimit is preserved up to homotopy typ e

Basic work on homotopy colimits has b een done by Segal Bouseld Kan

tom Dieck Vogt and Dwyer Kan See Hollender Vogt for a

recent survey

Two key results in this setting are the Pro jection Lemma XI Iiv which

sometimes allows one to replace colimits by homotopy colimits and the Homotopy Lem

ma XI I which compares the homotopy typ es of diagrams over the same

smal l category These to ols have found striking applications for example in the study of

subspace arrangements

The choice of contents for our to olkit for the manipulation of homotopy colimits is

partially motivated by the usefulness of corresp onding lemmas in the sp ecial case of order

complexes the discrete case when the spaces of the diagram are p oints In this case

there is a solid amount of theory available which has proved to b e extremely p owerful

and useful in quite diverse situations The key result is the Quillen Fiber Theorem

Quillens Theorem A see b elow All other basic to ols of the homotopy theory

of p osets such as the crosscut theorem order homotopy theorem complementation

formulas etc can b e derived from it We refer to Bjorner for an excellent account of

the theory Sect for an extensive survey of applications Part I and for further

references

The homotopy limits have not found immediate applications in combinatorics and

discrete geometry so far and this is the reason why we restrict our attention to the case

of homotopy colimits Also note that many results ab out homotopy limits can b e derived

from the case of homotopy colimits by standard duality pro cedures see Bouseld Kan

XI I and Hollender Vogt Sect

We provide several applications of our metho ds to various areas within mathematics

As a rst application in the eld of top ological combinatorics we present a new pro of

of a result by Bjorner on the homotopy typ e of complexes which generalizes the

Homotopy Complementation Formula of Bjorner Walker a to ol which has

proved to b e very p owerful in combinatorics Since this pro of aords the application of

many of the techniques provided in this pap er we give a detailed exp osition of it here

Then we present a new view of toric varieties Namely we start with the observation

that toric varieties are homeomorphic to homotopy colimits over the face p oset of the

fan dening the variety This immediately leads to a sp ectral sequence to compute the

homology of toric varieties isomorphic to one already employed by Danilov and to the

computation of its rational cohomology

We derive a new combinatorial formula for the homotopy typ es of quite general

arrangements such as Grassmannian arrangements that are asso ciated to linear sub

space arrangements by suitable functorial constructions More briey we cover two ap

plications for which details are contained in other pap ers We describ e a new result on

the homotopy typ e of the order complex of the p oset S G of nontrivial psubgroups of

p

a nite group G and we review results obtained by homotopy limit metho ds on the

top ology of subspace arrangements in and provide the equivalence with the results

of Vassiliev

Fundamental Concepts and Constructions

Basic Denitions and Motivating Examples of Diagrams

In the following all categories are smal l so their ob jects and morphisms form sets Any

partially ordered set can b e considered as a category with morphisms p ointing down

that is for x y P there is a unique arrow x y if and only if x y

A diagram of spaces over a small category A is a covariant functor F A T op into

the category T op of top ological spaces We denote for an ob ject a Ob jA the image

under F by F or by F a and for a morphism g a b MorA the image F g by

a

f or by F g If there is a unique morphism g a b in A b etween a and b then

g ab

we write f for f A morphism F X Y of diagrams X A T op and

ab g ab

Y B T op is a functor F A B together with a natural transformation from

X to Y F F Y Given a diagram F the homotopy colimit ho colimF is a space

asso ciated to F by a homotopy mixing construction see Section Before we pro ceed

with a reasonably detailed outline of the theory we intro duce several motivating examples

of diagrams of spaces

Constant Diagram For a top ological space X the constant diagram X is dened

A

by sending each ob ject of A to the space X and each morphism to the identity

id X X Of particular interest is the case when X fg is the onep oint

space In this case the constant diagram PT leads via homotopy colimits to the

A

construction of the classifying space BA of the category A In the sp ecial case when

A P is a partially ordered set a poset for short the classifying space BP will

b e seen to coincide with the order complex P of P

Group Diagram Given a discrete group G let A b e the category which consists of

G

one single ob ject and a morphism for each element g G Then the classifying space

BG of the group G Its universal cover of this category is the K G space BA

G

EG is constructed in a similar way from the category whose ob jects are the elements

of the group Here for each pair of elements g h G the unique morphism from g

to h is given by hg Finally any A diagram X A T op can b e interpreted

G G

as a Gspace X and it turns out that ho colim X EG X

A G

G

Subspace Diagrams Let U fX g b e a collection of subspaces of a top ological

i iI

space X The intersection poset P of the family U is the partially ordered set of

U

T

all nonempty intersections X J I ordered by reversed inclusion Regard

i

iJ

P as a small category Then the subspace diagram asso ciated to U is the diagram

U

D P T op sending each element of P to the corresp onding intersection

U U U

T

X with inclusions as morphisms This class of examples can b e used to see

i

iJ

many results from top ological combinatorics from a higher p ersp ective eg Borsuks

Nerve theorem

Arrangements as Diagrams An esp ecially interesting class of examples arises if X

in the previous example is a linear space ane space sphere pro jective space

and the family U is an ane spherical pro jective subspace arrangement The

subspace diagrams that arise this way have b een successfully used to deduce b oth

new and old results ab out the homotopy and homology of these arrangements and

their complements see Ziegler Zivaljevic Schap er

Orbit Diagrams Let G b e a Lie group acting on a space X with nitely many orbit

typ es From this arises a natural diagram O O T op over the category O of

G G

H

all Gorbits dened by O GH X see Recall that the orbit category O

G

H Gg and a morphism is dened as the category with Ob j O fGH j H

G

GH GQ for every inclusion H Q Generally any diagram over the orbit

category O will b e referred to as an orbit diagram

G

Toric Diagrams A toric diagram D is a diagram for which each space D is the stan

d

j

dard torus T S S j and each morphism d D D is a standard

ab a b

algebraic homomorphism i e a homomorphism that arises from an integer matrix

i j

K R R i dimD j dimD An imp ortant observation is that any

a b

compact complex toric variety can b e interpreted as the homotopy colimit of a

toric diagram over the face p oset of a complete fan see Section

In order to avoid pathological b ehavior of the top ological spaces involved and b ecause

this setting covers all application we can think of in our combinatorial setting we

assume that all spaces are compactly generated For the same reason we restrict our

attention to small categories which are not top ologized Note however that the general

theory see can b e develop ed for the case of top ological categories A for which the

map Ob jA MorA is a cobration

Limit Space Constructions

We will now discuss several constructions of limit spaces from a diagram D For that

we recall the notion of a simplicial space and its geometric realization The references

provide more information ab out simplicial sets and spaces their geometric realizations

and other general categorical constructions used in this pap er Nevertheless we try to b e

as selfcontained as p ossible

The category O r d has as ob jects the nite ordinals n f ng n its

morphisms are the nondecreasing maps f n m ie f i f i The set

i i

of morphisms of O r d is generated p I I by the maps and where

n n

i

n n is the increasing injection that do es not assume the value i for

n

i

i n and n n is the nondecreasing surjection that assumes the value

n

i twice for i n

A simplicial space is a contravariant functor F O r d T op A simplicial set

is a contravariant functor F O r d S et We will not distinguish b etween simpli

cial sets and simplicial spaces equipp ed with the discrete top ology S impl icial S ets and

S impl icial S paces are categories with the obvious morphisms that is whose morphisms

are the natural transformations b etween functors O r d S et resp O r d T op

There is a simple geometric realization functor R O r d T op that asso ciates with

n the standard ndimensional simplex

n

n o

X

n

t t t R t t t t

n n n i

i

i i i

and asso ciates with and the face op erators R resp the degeneracy op erators

n n n

i

R Viewing O r d as the category of nite chains and order preserving maps the

n

functor R asso ciates with chains their order complexes and with order preserving maps

the corresp onding simplicial maps

Now we are able to review some imp ortant limit space or total space constructions

arising from categories and diagrams

Geometric Realization Given a simplicial space F one denes its geometric realiza

i i

tion kFk Denote by F Fn the image of n under F by d the image of

n

n n

i i

under F and by s the image of under F Then kFk is obtained as the quotient

n n

U

of the disjoint union F mo dulo the equivalence relation dened by

n n

n

i i

d x p x R p for x F and p

n n

n n

i i

s x p x R p for x F and p

n n

n n

With this geometric realization k k is a functor from S impl icial S paces to T op

Classifying Spaces If A is any small category then there is a natural simplicial set

F O r d S et asso ciated to it Namely the image of n under F consists of all

A A

f f

n

1

sequences of ob jects Ob jA and maps f Mor in

n i i A i i

i

the category A The maps d are given by

n

f f f

n

2

i

i

i i n

f f

n

1

f f

f f

i

i+1 i

n

1

d

n

i n

n

n i i

f

f f

n1

1

i

i n

n i i

i

The maps s are given by

n

id

f

f f f f f

i+1

n n

1 1

i

i

i

s

i n n i

n

The geometric realization of the simplicial set F is the classifying space B A of

A

the category A

colimD The colimit of a diagram D is the top ological space obtained from the direct

U

sum D mo dulo the equivalence relation generated by x y for x D

a a

aA

y D if d x y for some g a b We write colimD for the colimit of the

b g ab

diagram D

ho colimD The homotopy colimit of a diagram D is dened as the quotient of the direct

U

sum B A D by an equivalence relation Here A denotes for a given

a a a

aA

category A and a Ob j A the category of all arrows a b emanating from a with

commutative triangles as morphisms b etween ob jects a b and a b If A P

is a partially ordered set then A P fp P j p ag The equivalence

a a

relation is dened as follows For all morphisms f a b consider the maps

B A D B A D

b a b b

p x p d x

f ab

B A D B A D

b a a a

p x B f p x

Then is the transitive closure of p x p x

If it is imp ortant to emphasize the indexing category A then the homotopy colimit

of a diagram D is denoted by ho colim D

A

Bar Construction A very general construction of a limit space is a construction given

op

by Hollender Vogt Given diagrams D A T op and E A T op

op

over a category A and its opp osite category A there is an asso ciated total space

B D A E Its construction generalizes all limit space constructions presented so

far except for the colimit construction To the data D and E one asso ciates a

simplicial set D A E The space of nsimplices of D A E is the space of all

threads

f

f f

n1

n

1

x y

n n

More precisely this space can b e describ ed as the and y E where x D

n

0

U

space D E where the direct sum is taken over all nchains

f

f

n

n

1

n

0

f f

n

1

i

of morphisms in A The maps d i n are given by

n

n

f f f

n

2

i

e y i x

i i f n

1

f f

n

1

f f

f f

i

i+1 i

n

1

y d x

n

y i n x

n

i i n

f

f f

n1

1

i

x y i n d

n i i f

n

i

The maps s i n are given by

n

id

f

f f f f

i+1

n n

1 1

i

i

y x s x y

i n n i

n

Denote by B D A E the geometric realization of the simplicial space D A E The

homotopy colimit and the classifying space are sp ecial cases of the geometric realization

of simplicial spaces of the form D A E Namely let PT b e the diagram over the

A

category A that assigns to each ob ject a in A the onep oint space fg Then the clas

op

sifying space of the category A is the geometric realization of PT A PT For

A A

a diagram D A T op the homotopy colimit ho colimD is homeomorphic to the space

op

B PT A D

A

We may observe that for every diagram D over a category that is a p oset P there

U

is a canonical partial order on P D dened by setting x x for x D and

p p

pP

0 0

x x With this denition there is a canonical if and only if p p and d x D

pp p

bijection P ho colimD however the top ology on the two spaces is dierent in

fact with the usual top ology on simplicial complexes the subspaces D P get a

p

discrete top ology Note however see Section that there exists a convenient geometric

mo del for ho colimD realizing this space as a subspace of the join J D D

pP p

This construction can b e seen as a natural extension of the order complex construction

to diagrams over p osets

Examples Here are some simple examples for the construction of homotopy colimits

of diagrams over p osets

i If D is a onep oint space for all p P i e D PT then ho colimD is isomorphic

p

to the order complex of P

ii If P fpg is a oneelement p oset then ho colimD D

p

0

D is the mapping iii If P fp p g has two p oints and p p then ho colimD D

f p p

0 0

D cylinder of the map f d D

p p p p

The mapping cylinder is homotopy equivalent to the image space D which is the

p

colimit of the diagram

0

is the disjoint iv If P fp p g and p p are not comparable then ho colimD D D

p p

union

In this case it agrees with the colimit colimD of the diagram

v Let P fp p p g b e the threeelement p oset with p p p p for which

00

fg a p and p are incomparable and let D b e a diagram over P with D

p

onep oint space Then the homotopy colimit ho colimD is homeomorphic to the

0 0

D D mapping cone of the map d

p p pp

vi If D is a diagram such that all the spaces D are identical and the maps are identity

p

P D maps then ho colimD

p

The colimit of such a diagram is D if P is connected

p

Simplicial Homotopy Lemma and the Gluing Lemma

Denition A simplicial space F O r d T op is called good if for any surjective map

n n in O r d the image of F F n F n is a closed cobration

Prop osition Simplicial Homotopy Lemma

Let A B be two good simplicial spaces Let F A B be a map of simplicial spaces If

F A B is a homotopy equivalence for al l n then the induced map

n n n

kFk kAk kBk

is a homotopy equivalence

Go o d simplicial spaces were dened by Segal App endix under the name prop er

simplicial spaces In and in the closely related alb eit not coinciding classes

of prop er and strictly prop er spaces are intro duced Prop osition is known as the

MayTornehave or the MayTornehaveSegal theorem Pro ofs can b e found in either of

the references mentioned ab ove

The Gluing lemma is a key result It has rep eatedly b een used in inductive pro ofs

of results establishing that two spaces are homotopy equivalent a notable example is the

pro of of Prop osition This lemma is a basic example for an imp ortant general prin

ciple which informally says that a lo cal homotopy equivalence is also a global homotopy

equivalence see the Homotopy lemma

Lemma Gluing lemma

Consider the commutative diagram of spaces

f f

2 1

C A B

h h h

1 0 2

g g

1 2

V U W

Assume that h h h are homotopy equivalences and that either both f and g are co

brations or both f and g are cobrations Let X B C resp Y V W denote

A U

the colimit or pushout of the diagram formed by the maps f f resp g g Then

the induced map h X Y is also a homotopy equivalence

Comparison Results for Diagrams of Spaces

In this section we set up the to olkit for applications of diagrams of spaces In the subse

quent section we will show how these very general results sp ecialize to forms applicable in

various combinatorial situations where the emphasis is on diagrams over lo cally nite

p osets The to olkit consists of a sequence of prop ositions and lemmas most of which

are part of standard to ols of an algebraic top ologist These results allow us to recognize

a top ological space as the homotopy colimit of a diagram Projection Lemma to

check if a given morphism F X Y of diagrams induces a homotopy equivalence

ho colim X ho colim Y of the corresp onding homotopy colimits Homotopy

A B

Lemma to change the underlying small category of the diagram without changing the

homotopy typ e of the homotopy colimit to compute or to nd a standard combinato

rial form of the space ho colim X Wedge Lemma etc

A

Pro jection wedge and the quasibration lemma

Prop osition Projection Lemma

Let X C T op be a diagram such that X X a X b is a closed cobration

f ab

for any morphism f Mora b a b O bj C Then the natural map

ho colim X colim X

C C

is a homotopy equivalence

The next result that we are heading for the Wedge Lemma has b een formulated

and used b efore in the case of diagrams over p osets In order to formulate and prove

a more general form of this lemma we need two preliminary denitions

Denition A diagram X A T op is called a diagram with constant maps if

O bj A and any nonidentity morphism f a a f id the map for all a a

a

X f X a X a is a constant map A special diagram with constant maps is a

diagram with constant maps with the additional prop erty that for any ob ject a O bj A

there exists a p oint x X a such that for any morphism f b a f id the

a a

one element set ImX f is precisely fx g In addition it is assumed that all spaces

a

X a are well p ointed which means that X a fx g is a cobration pair If the

a

category A has an initial ob ject a ie an ob ject such that for each a O bj A there is a

unique morphism from a to a then a diagram with constant maps resp sp ecial diagram

with constant maps over A is called an initial diagram with constant maps resp sp ecial

diagram with constant maps

Denition Given a category A and a O bj A the undercategory A often de

a

U

noted by anA or aI d is the category with ob jects O bj A Mora b and

a

bO bj A

f g

an arrow from a x to a y for each commutative triangle formed by f g and a mor

h

phism x y g h f Given c O bj A the truncated category obtained by deleting

c and all incident morphisms is denoted by Annc If we delete from A the initial ob ject

a

id

a a and all incident morphisms we obtain the truncated undercategory A nnid which

a a

is denoted by A Sp ecially if A is a p oset category P then A and A are p oset

a a a

categories P and P

a a

Lemma Let X be an initial diagram with constant maps over a category A with a

as an initial object Then the homotopy colimit of this diagram has the fol lowing join

decomposition

ho colim X X a B Anna

A

where Anna is the truncated category dened above and B A nna its classifying space

Moreover if X a is not a singleton these spaces are homeomorphic

Pro of For an ob ject x O bj A let f Mora x b e the unique morphism connecting

x

a and x eg f id If g Morx a then g f id hence X g X f id If

a a x a x

X a

x a this is p ossible only if X a consists of a single element in this case the formula is

correct since b oth sides of the equation are contractible spaces If jX aj then the only

morphisms with a as the domain or co domain are of the form f for some x O bj A

x

X a B A nna follows by denition In this case the formula ho colim X

A

Remark There are interesting examples of diagrams X A T op see and Sec

tion which have the prop erty that for all a b O bj A and f a b f id

a

the map X f X a X b is homotopically trivial Typically these diagrams can b e

transformed usually with the aid of the Homotopy lemma Prop osition into sp e

cial diagrams with constant maps without aecting the homotopy typ e of the homotopy

colimit The following lemma shows that in these cases we can actually compute these

homotopy typ es

Prop osition Wedge lemma

Suppose that X is a special diagram with constant maps over A If the category A is

connected then the homotopy colimit of X has the wedge decomposition

ho colim X B A X a B A

A a

aO bj A

If the category A is not connected then the formula applies to each of the connected

components

Pro of Let X PT b e the canonical collapsing map from X to the constant

diagram PT The assumption ab out p oints x X a leads to a morphism of diagrams

a

PT X which is a right inverse to I d hence B A is a retract of ho colim X

A

For each a O bj A let F A A b e the naturally dened functor There are three

a a

diagrams over A the constant diagram Z a A T op b X a the pullback

a a

X cf Denition and the sp ecial diagram with constant maps diagram Y a F

a

X a obtained from Y a by collapsing to a p oint all spaces except for the space X a that

is asso ciated to the ob ject id It is clear that ho colim Z a B A X a and that

a A a

#a

r

a

there is an obvious retraction B A X a ho colim X a By the denition of the

A

#a

homotopy colimit

ho colim X B A X a

A a

aO bj A

so there is a map f B A X a ho colim X which factors as follows

a a A

r s

a a

B A X a ho colim X a ho colim X

a A A

#a

All identications in o ccur along the space B A and we observe that ho colim X is

A

obtained by simultaneous adjunction pushout for all a O bj A of the following form

j g

a a

ho colim X a B A B A

A a

#a

where j is an inclusion map and g is the map induced by the functor F A A

a a a a

The space B A is contractible so we can according to the Gluing lemma Lemma

a

replace maps j and g in by constant maps without changing the homotopy typ e

a a

of the adjunction space This can b e done simultaneously for all a O bj A which

together with Lemma and the observation that A nnid A leads to the desired

a a a

formula

The obvious collapsing map of diagrams X PT where X is an arbitrary A

A

diagram and PT is the constant p ointdiagram over A induces a map ho colim X

A A

B A Under certain strong conditions this map turns out to b e a quasibration a

concept intro duced by Dold Thom A map f X Y b etween top ological spaces is

a quasibration if for any y Y and x f y the induced maps f X f y x

i

Y y are isomorphisms for all i The following prop osition is essentially due to

i

Quillen

Prop osition Quasifibration Lemma

If X is an Adiagram such that al l the maps X f X a X b f Mor a b

A

are homotopy equivalences then the natural projection map ho colim X B A is a

A

quasibration In particular if B A is contractible then for every a O bj A the map

X a ho colim X is a homotopy equivalence

A

From lo cal to global homotopy equivalences

This is the rst circle of results ab out homotopies b etween homotopy colimits of diagrams

Prop osition Homotopy Lemma

Suppose that X and Y are diagrams over C and id X Y is a morphism such

that c X c Y c is a weak homotopy equivalence for al l c O bj C Then the

induced map

ho colim X ho colim Y

C C

is also a weak homotopy equivalence

This Homotopy Lemma is a central result ab out diagrams of spaces As noted ab ove

it has evolved from the Gluing lemma Prop osition in the pap ers of tom Dieck

Bouseld Kan XI I Vogt and others Both the Homotopy lemma and

the following more general result from for the bar construction as discussed in

Section can b e deduced from the results of Section

op

Prop osition Let X X C T op and Y Y C T op be diagrams Suppose

id X X and id Y Y are morphisms of diagrams such that c

X c X c and c Y c Y c are homotopy equivalences for al l c O bj C

Then the induced map

C X B id B Y C X B Y

is also a homotopy equivalence

The Homotopy Lemma is a very convenient to ol if we want to compare homotopy

typ es of homotopy colimits of two diagrams over the same category C For comparison

of homotopy colimits of diagrams over dierent categories we have at our disp osal the

following two elegant results They b oth describ e p ossible changes of the indexing category

of a diagram that do not aect the homotopy typ e of the homotopy colimit

Denition Given a functor F C D the pul lback functor F asso ciates a C

diagram with every D diagram It is dened by F Y Y F for any D diagram

Y If d Ob jD let dF b e the category which has as ob jects all arrows of the form

f f

d F c for some c Ob jC and for morphisms b etween arrows d F c and d F c

commutative triangles F h f g where h Mor c c If F I d is the identity functor

C

we recover the denition of undercategory from Denition

Prop osition Cofinality Theorem XI

If a functor F C D is right conal ie if the classifying space B d F is homo

topical ly trivial for al l d D then the induced map

ho colim F Y ho colim Y

C D

is a homotopy equivalence

The Segals homotopy pushdown functor F asso ciates a D diagram to any C

h

diagram X C T op

op

Denition For every category D there is a tautological D D diagram D

op

D D T op of discrete spaces dened by d d Mor d d If d Ob jD is

D

xed then let D D T op b e the D diagram dened by x Mor d x Similarly

D

d

op

for a xed d Ob jD the D diagram dened by x Mor x d is denoted by

D

op

0

D Given a functor F C D and d Ob j D the D diagram D is via F

d d

op

seen also as a C diagram so the space B D C X is well dened The assignment

d

d B D C X denes a D diagram F X which is called the homotopy pushdown

d h

of X or the left homotopy Kan extension along F

Prop osition Homotopy Pushdown Theorem

Given a functor F C D and a C diagram X there is a natural homotopy equivalence

ho colim F X ho colim X

D h C

Homotopies Arising From Natural Transformations

A prop osition of Segal Prop says that any natural transformation of functors

F F C D induces a homotopy on the level of geometric realizations This principle

has numerous consequences and applications some of them are collected in this section

Recall that a morphism F X Y of diagrams X A T op and Y B T op

is a functor F A B and a natural transformation from X to Y F The result of

Segal deals with top ological categories and not with diagrams However the connection

with results quoted here b ecomes transparent if a diagram X A T op is viewed as a

U

f

top ological category with X a as the space of ob jects and an arrow x y for

aO bj A

each pair of ob jects x y and a morphism f Mor a b such that x X a y X b

A

and X f x y We are indebted to the referee for the following prop osition and its

corollaries The referee also p ointed out that these results can b e immediately deduced

from Prop osition in however in the more explicit form presented here they are

closer to our intended combinatorial applications

Prop osition Suppose that X A T op and Y B T op are diagrams of spaces

over A and B Assume that F X Y and G Y X are morphisms of these

diagrams such that either

there exists a natural transformation G F I d such that X id or

a natural transformation I d G F such that X

X

A T op

F id

B T op

Y

G id

A T op

X

Then F and G induce continuous maps f ho colim X ho colim Y and g

A B

ho colim Y ho colim Y such that g is a left homotopy inverse to f

B A

g f I d

Moreover if there exists a morphism of diagrams H X Y and either a natural

transformation H G I d such that Y id or a natural transformation

I d H G such that Y then F G and H induce homotopy

equivalences

ho colim X ho colim Y

A B

Corollary Let i B A be an inclusion of B as a ful l subcategory of A F A B

a functor which maps B to B and I d i F a natural transformation Let X be an

A

Adiagram and let X jB denote the restriction of X to B Then the canonical inclusion

map induces a homotopy equivalence

ho colim X jB ho colim X

B A

Pro of The inclusion of diagrams X jB X induces an inclusion map h

ho colim X jB ho colim X The natural transformation denes a map X X jB

B A

of diagrams a F a X a X a which induces a map h ho colim X

a A

ho colim X jB Then h is b oth the right and left homotopy inverse of h Indeed

B

this follows from Prop osition with F H F X G i id

Y X jB X i and i

Corollary Let F A B be a left adjoint to G B A Let X be an Adiagram

Then G id is a map of diagrams which induces a homotopy equivalence

ho colim X G ho colim X

B A

Pro of Let I d G F and F G I d b e the adjunction transformations Apply

Prop osition with F H F Y X G G G id I d

G F and F G I d

Diagrams over p osets

Diagrams over p osets play a sp ecial role in this pap er since the ma jority of diagrams

arising in combinatorics are of this typ e The situation resembles the use of diagrams

of spaces in the theory of transformation groups where this technique is also of utmost

imp ortance A top ologist primarily interested in group actions may nd it convenient

to have the key statements formulated in the language of Gspaces although they are

consequences of more general results An example is the result that an equivariant map

H H H

f X Y induces a homotopy equivalence f EG X EG Y if f X Y

G G

is a homotopy equivalence for any subgroup H G

A similar tendency exists in combinatorics For example the reference where

Prop ositions and are formulated deals only with p osets despite the fact that

Quillen himself was in p ossession of much more general statements his well known The

orems A and B All these results are relatives of theorems formulated in Sections

and and they played an imp ortant role in the development of the general theory

This is the rst reason why we collect some of the corollaries of the general results in

a separate section A second reason is that there exist results which hold for p osets and

which do not have useful analogs for general categories

Homotopy Colimits Revisited

The homotopy colimit construction is supp osed to b e a direct extension of the order

complex construction for lo cally nite p osets P It is shown in Section that the join

J X X X of a family X fX g of spaces has a simple description

n i in

as the homotopy colimit of a natural diagram of spaces Here we note that conversely at

least in the case of p osets the homotopy colimit can b e describ ed as a subspace of a join

of spaces

Prop osition Let D P T op be a diagram over a local ly nite poset P Let

J D D be the join of al l spaces in this diagram and let

pP p

n o

D t x t x J D j x D p p d x x

m m i p m p p j j

i j +1 j

Then this space is natural ly homeomorphic to the homotopy colimit of D

ho colim D D

P

This characterization of homotopy colimits generalizes the order complex construction

PT where PT is the constant diagram asso ciated to P as in since P

P P

Section

Comparison Results for Diagrams Over Posets

In this section we briey review some of the most useful comparison lemmas which are

meaningful only for order complexes of p osets Examples of such results are the Cross

cut theorem and the Homotopy Complementation Formula the latter one b eing

a sp ecial case of Theorem These result have found many interesting combinatorial

applications Also by showing some of the most useful corollaries of general results from

Section we provide a link b etween these results and the general theory of diagrams

Quillens Theorem A or the Fiber Theorem can b e seen as a corollary and a result

that motivates b oth the Cofinality Theorem Prop osition and Segals Push

down Theorem Prop osition Quillens Theorem B or the Order Homotopy

Theorem is a corollary and a go o d illustration of results from Section

Prop osition Quillen Fiber Theorem Quillen Walker

If f P Q is a map of posets such that the order complex of f Q is contractible

q

for al l q Q then the simplicial map f P Q is a homotopy equivalence

Similarly if the order complex of f Q is contractible for al l q Q then f is a

q

homotopy equivalence

Prop osition Order Homotopy Theorem Quillen see Bjorner

If f P P is a decreasing poset map that is f p p for al l p P then P is

homotopy equivalent to f P

A join semilattice is a p oset P such that every nite subset has a unique minimal

upp er b ound If P is of nite length then this implies that P has a unique maximal

element The crosscut complex P of a join semilattice is the simplicial complex of

all nonempty subsets of minP that have an upp er b ound in P n

Prop osition Crosscut Theorem see Bjorner

For every join semilattice P of nite length the crosscut complex P is homotopy

equivalent to the order complex of P n

Next we come to the combinatorial version of the Projection Lemma The nat

ural combinatorial setting for its application see Sections and is the following

A top ological space X is covered by a nite set X of closed subspaces The inter

i iI

S T

section poset P of the covering X X is dened on the set of all spaces X

i j

iI j J

J I that o ccur as an intersection of spaces X with the reversed inclusion as its

i iI

order relation There is a natural diagram D P T op asso ciated to the covering

Namely to each element p P one assigns the corresp onding intersection D and denes

p

the maps as the natural inclusions

Lemma Projection Lemma Segal Bouseld Kan XI Iiv

S

Let X X be a covering of the space X by a nite set of closed subspaces Let P

i

iI

be the intersection poset P and let D be the corresponding P diagram Then the natural

map ho colimD colim D X induces a homotopy equivalence ho colimD X

In general a standard application of the homotopy colimit to ols pro ceeds from a

diagram D asso ciated to a covering to another more simple diagram by suitably mo dify

ing D using a p oset version of the Homotopy Lemma the Cofinality Theorem

or the Upper Fiber Lemma

Lemma Homotopy Lemma

Let D and E both be P diagrams and assume that there is a map of diagrams id

D E such that D E is a homotopy equivalence for each p P then induces

p p p

a homotopy equivalence b ho colimD ho colimE

If one wants to change the p oset underlying a diagram on needs an analog of the

Confinality Theorem

Lemma Inverse image lemma

Let f P Q be a poset morphism and let E be a Qdiagram Dene the P diagram

0 0

e f E by f E E and f e

f pf p p f p pp

If for al l q Q f Q is contractible then f induces a homotopy equivalence

q

ho colimf E ho colimE

In contrast to the preceding lemmas that are just reformulations of the general state

ments for small categories in the situation of diagrams over p oset the Upper Fiber

Lemma do es not follow immediately from the general to ols presented here Therefore we

provide a pro of

Lemma Upper Fiber Lemma

Let D be a P diagram and let E be a Qdiagram Assume f D E is a map of

diagrams If f induces a homotopy equivalence of the restrictions

b ho colimD j ho colimE j for al l q Q

pP f pq Q

q

then f induces a homotopy equivalence b ho colimD ho colimE

Pro of We pro ceed by induction on the cardinality jQj of Q Let q Q b e a minimal

element of Q Then Q Q Q for Q Q and Q Qnfq g The p oset Q Q Q

q

equals Q We set P f Q for i which implies P P P and

q i i

P P P

If q is the unique minimal element of Q ie Q Q then the assertion follows triv

ially from the assumptions Otherwise induces a homotopy equivalence b holim D j

P

i

ho colimE j for i by induction Since ho colimD j ho colimD j ho colimD j

Q P P P

1 2 3

i

and ho colimD j ho colimD j ho colimD j it follows from the Gluing Lemma

Q Q Q

1 2 3

that induces a homotopy equivalence b etween ho colimD ho colimD j ho colimD j

P P

1 2

and ho colimE holim E j ho colimE j

Q Q

1 2

If one is lucky and surprisingly often one is indeed lucky one is able to transform

a diagram D into a diagram that allows an application of the Wedge Lemma whose

p oset version we state next

Lemma Wedge lemma

Let P be a poset with maximal element Let D be a P diagram so that there exist points

0 0 0 0 0

for p p Then x c is the constant map d for al l p such that d D c

p pp pp p p

ho colimD P D

p p

pP

where the wedge is formed by identifying c P D with p P D for

p p p

p

We close this to olset of lemmas by a result that is a simple consequence of the denition

of a homotopy colimit

Lemma Embedding lemma

Let id D E be a map of P diagrams If D E is a closed embedding for

p p p

every p P then induces a closed embedding

b ho colimD ho colim E

In particular if D is a P diagram and Q P is a subposet then ho colimD jQ ho colimD

is a closed embedding

Applications

Combinatorial Ob jects as Homotopy Colimits

In this section we show how some well known combinatorial constructions can b e usefully

related to diagrams and their homotopy colimits

n

Prop osition Let X fX g be a col lection of spaces Denote by B the poset of

i n

i

fng

al l nonempty faces of an nsimplex ie the poset n fg ordered by inclusion

Let D B T op be the fol lowing diagram associated to the col lection X For A B

X n n

Q

let D A X and for A B let d D A D B be the canonical

X i AB X X

iA

projection Then the join J X of al l spaces in the family X is natural ly homeomorphic

to the homotopy colimit of this diagram

D ho colim J X X X X

X B n

n

Figure The join of S and S as a homotopy colimit

The formula given in Prop osition is a very simple and useful representation of the

join of spaces exp osing much of the underlying combinatorial structure

An imp ortant variation of the diagram ab ove arises if X S for all i This is a

i

toric diagram in the sense of Section There is a natural diagonal action of the circle

jAj

group U on all tori D A S Dividing by this action leads to a new diagram

E D U and a well known fact is that this leads to a combinatorial description of

n

the complex pro jective space C P

n

C P E ho colim

B

n

A very useful concept which has found numerous applications in combinatorics is the

deleted join operation Given a simplicial complex K the pth deleted join

p

p

K is the sub complex of the join K K K of p copies of K which consists of all

p

simplices K which have the form where i p are pairwise

p i

vertex disjoint simplices of K Some very imp ortant complexes can b e constructed by

iterating the join and the deleted join op eration for example the well known chessb oard

complexes arise this way The simplest complex of this form is J

pq N q

q

N N

where is the standard N dimensional simplex This space viewed as a

space with the obvious cyclic Z action has b een applied in combinatorics to problems

q

of Tverb erg typ e to the necklace partition problem etc see Motivated by these

and other similar examples it is natural to lo ok for homotopy colimit representations for

deleted joins as well

p

There are two obvious ways to asso ciate a diagram to the space of the form K

p

p

One can view K as a subspace of K which has the homotopy colimit representation

describ ed in Prop osition The sub diagram which arises this way is similar except that

j

j

the pro ducts K are replaced by deleted products K j p Recall that the deleted

j

j

pro duct K is a cell sub complex of the cell complex K consisting of all cells of the form

where k j are pairwise vertex disjoint simplices in K

j k

p

Another p ossibility is to link K with a diagram over the face p oset of K itself via the

p

collapsing map K K The diagram arising this way is a nite space diagram For

q

N

example if K the space K J is the homotopy colimit of a diagram of the

N q

typ e describ ed in Prop osition where X q Z for all i N These

i q

and other examples seem to indicate that it may b e useful to study discrete analogs of

j j

toric diagrams with the discrete tori Z Z Z in place of T S S

q q

q

Sp ecially it is an interesting question which convex p olytop es admit natural discrete toric

diagrams

Bjorners generalized homotopy complementation formula

Bjorners generalized homotopy complementation formula is an eective to ol to com

pute the homotopy typ e of a simplicial complex in the case when a large contractible

is known whose connections to the rest of the complex are not induced sub complex

A

to o complicated

In the following we provide a diagrammatic pro of of Bjorners result thereby

demonstrating the applicability of some of our lemmas

S

Let b e a nite abstract simplicial complex with vertex set S let A S b e

a subset of its vertex set and denote by A the complement S nA of A

A

Let f Ag b e the induced sub complex on A and similarly

A

A In the following the key assumption we will make is that the induced complex on

A

is contractible

A

Theorem Generalized Homotopy Complementation Formula Bjorner

S

For any simplicial complex and A S dene a new simplicial complex T by

A

taking the union of al l the simplicial complexes

for p star

A

where p is an additional point p S and denotes the join of two complexes

If is contractible then and T are homotopy equivalent

A

A

Pro of Let P b e the p oset of all nonempty faces of ordered by inclusion and let

Q f P P g

partially ordered by the condition

IntP Thus Q is isomorphic to the p oset of all intervals in P ordered by inclusion Q

For P we dene

D star

A

Then we clearly have inclusion maps d D D whenever This denes a

P diagram D with

colim D ho colimD

where the equality holds by denition of the colimit this is a subspace diagram and

the homotopy equivalence is an application of the Projection Lemma

Similarly for Q we dene

E p star

A

0 0 0 0

Then we get inclusion maps e E E whenever This

denes a Qdiagram E with

T colimE ho colimE

A

by denition of the colimit and the Projection Lemma

Thus the claim of the theorem is reduced to proving that the homotopy colimits of

the P diagram D and the Qdiagram E are homotopy equivalent This demands use of

our new homotopy lemmas since the p osets P and Q are quite dierent

The canonical p oset map to use is

f Q P

This f induces a homotopy equivalence b etween the p osets P and Q Walker To see

this we observe that the lower b ers f P are canonically isomorphic to Int P

On the p oset IntP we have an increasing map g IntP Int P given by

Thus by the Order Homotopy Theorem the b er Int P is

P which is a cone homotopy equivalent to the image g IntP

Now we mo dify E and D a little We dene a new Qdiagram E whose spaces are

E star p

A

and whose maps are the obvious inclusions Furthermore there is a map E E

which is the identity map on Q and b etween the spaces uses the inclusion maps

p star p star star p

A A A

is contractible Thus by the Homo which are clearly homotopy equivalences since

A

topy Lemma induces a homotopy equivalence

b

ho colimE ho colimE

Similarly we dene a new P diagram D whose spaces are

star p D

A

and whose maps are the obvious inclusions Furthermore there is a map D D

which is the identity map on P and b etween the spaces uses the inclusion maps

star star p

A A

which are clearly homotopy equivalences Thus by the Homotopy Lemma induces

a homotopy equivalence

b

ho colimD ho colimD

Finally at this stage we see that E is an inverse image diagram E f D where

f is a p oset map whose lower b ers we have already checked to b e contractible Hence

the Inverse Image Lemma implies a homotopy equivalence

ho colimE ho colimD

which completes the pro of

Alternatively one could derive ho colimE ho colimD also from the Upper Fiber

Lemma together with the Projection Lemma since E and D are subspace

diagrams as well

We note that there are alternative ways to describ e the construction of T For

A

example one can as Bjorner do es in his manuscript start from a wedge or a disjoint

union of the spaces E p star and then check that all the

A

identications of the colimit colim E are generated by identifying for the identical

sub complexes p star in E and in E

A

Also there are countless variations p ossible corresp onding to dierent coverings of

the complex The b eauty of Bjorners setup is that his transformations of end up

with a subspace diagram and thus with a colimit instead of a homotopy colimit which

leads to an eective mo del for It seems to us that the diagram techniques yield

A

an extremely natural and convenient setting for the pro of of the generalized homotopy

complementation formula and similar results

Toric Varieties

In this section we give a representation of the top ological space underlying a toric variety

see Danilov Fulton and Ewald for general background on toric varieties as

the homotopy colimit of a diagram For this we recall a description due to MacPherson

n

see Yavin Fischli of a toric variety A decomp osition of R into a complex

of closed convex p olyhedral cones with ap ex is called a complete fan If all cones

n

in are generated by lattice p oints in Z then is called rational Assume that is a

n

complete and rational fan in R Then let P b e the cell decomp osition of the unit ball

n

in R that is dual to the one induced by For we denote by the cell in P that

corresp onds to Thus is a cell of dimension n dim

n n n n

We identify the ntorus T with the image of the pro jection map R R Z For

all cones the image of under this pro jection is a subtorus span T

R

of T Since is rational this is a closed subtorus of dimension dim Thus the quotient

n

T T is a real torus of dimension n dim

n n

The toric variety X is obtained from P T by taking the quotient of T by

the action of T on T for each This leads to a nice compact Hausdor quotient

n

space since we take quotients by larger tori on T In particular we see that the

toric variety X has a welldened map X P for which the b er over any interior

p oint of is isomorphic to T T

Let P b e the p oset whose elements are in bijection with the cones in and whose

order relation is dened by reversed inclusion of the cones in Thus P is the p oset

of nonempty faces of P ordered by inclusion In particular P has a largest element

corresp onding to the dimensional cone fg We construct a diagram D over the p oset

P as follows For set D T T Top ologically D is an n dim torus

The map d for is the map induced by the pro jection T T T T

n

Prop osition Let be a complete and rational fan in R Then the toric variety X

is homeomorphic to the homotopy colimit of the diagram D associated with

X ho colimD

Pro of Let b e the map that sends T T P to its image in X By construction

U

of D the map is compatible with the equivalence relation on T T P

P

b b

Hence induces a map ho colimD X It is routine to check that is indeed a

homeomorphism

The resolution of singularities of a toric variety also ts our homotopy colimit frame

n

work Namely let b e two complete rational fans in R such that is a renement

in such that of ie for every op en cone in there is an op en cone

0

P Also assume that is a cone in whose Thus there is an induced map f P

interior is contained in the interior of the cone of Then the inclusion induces

0 0

T T Is is easily seen that induces a map of diagrams a surjective map T T

b b

0 0

X The map is ho colimD X Hence there is an induced map ho colimD

0

surjective since f and all are surjective

Prop osition Let be complete rational fan which is the renement of the complete

b

0

X rational fan Then there is a surjective map X

It is well known that X is nonsingular if and only if is simplicial ie all cones

n

are simplicial and unimo dular ie all fulldimensional cones are equivalent to fx R

n

x g under unimo dular transformations from GLR Z For an example that shows

that H X Q is not a combinatorial invariant of in general see It is also well

known that for any complete rational fan there is a simplicial and unimo dular complete

b

rational fan which renes In this case is a resolution of singularities

One can also use our results to investigate the cohomology of a toric variety For

this we set up a sp ectral sequence intro duced by Segal which uses the ltration of

ho colimD by the sskeleta of the order complexes For a simplicial complex we denote

s

by its sskeleton

s

Assume D is a P diagram for a p oset P Then we denote by ho colimD the image of

U

s

D P in ho colimD The ltration ho colimD ho colimD ho colimD

p p

pP

e

denes a sp ectral sequence with termination H ho colimD in the E term and E

st

s s

e

H ho colimD ho colimD Following Segals arguments one nds that E is given

st

st

M

s

e

Now assume c is a s tcell in ho colimD by H T T

s t

s

P

s

0

Then the dierential of the cell complex ho colimD is given by

s

X

i

c b c

s i s

i

s s

c c d

s s

s

s1

Thus the dierential d E E applied to the cell c equals

s

st st st

s

X

i s

b c d c where c

i s s

s

s1

i

is a cycle in H T T

t

s

From this it is easily seen that our sp ectral sequence is isomorphic to the deRahm

Ho dge sp ectral sequence applied by Danilov Chap x to compute the cohomology

of a toric variety

Subspace Arrangements

n

Arrangements of ane subspaces in R also allow an application of the homotopy col

n

b

imit metho d Let A b e a nite set of ane subspaces in R Let us denote by A the

n n

corresp onding arrangement of spheres in the onep oint compactication S of R Under

b

our assumptions intersections of spheres in A are again spheres or the compactication

p oint The following result can b e deduced from the Projection lemma the

Homotopy lemma and the Wedge lemma

Theorem Ziegler Zivaljevic

n

c

Let A be a nite set of ane subspaces in R Let U be the onepoint compactication

A

of the settheoretic union of the subspaces in A and let P be the intersection poset of A

Then

dimp

c

U S P

A p

pP

An equivalent result can b e found in Vassiliev I I I x Thm In Vassilievs for

mulation the spaces P are replaced by quotients of simplices by crosscut complexes

p

the spaces K p in his notation More precisely for an arbitrary subspace V corresp ond

ing to some p oint p p in the intersection lattice P of A let V V b e the subspaces

V t

in A such that V contains V as a subspace Let p b e the simplex which is spanned

i

in the abstract sense by the vertices V V Vassiliev calls a face of p marginal

t

if V is not the intersection of the subspaces corresp onding to the vertices of Thus the

marginal faces are the simplices in the crosscut complex P of P By the Cross

p p

cut Theorem the complex of marginal faces is homotopy equivalent to P In

p

dimp dimp

Vassilievs formula the spaces S P are replaced by S pP

p p

Let us analyze pP If P is the full simplex p then pP and by

p p p

dimp

the Crosscut Theorem also P are contractible In particular S P

p p

dimp

and S pP are contractible If P is some nonempty part of the

p p

b oundary of p then pP is the susp ension of P Thus again the Cross

p p

dimp dimp

cut Theorem shows that S P and S pP are homotopic If

p p

P is empty then we have to interpret pP as the susp ension of the empty

p p

space which is in our denition the join with a two p oint space Then the homotopy

equivalence also follows in this case

n

By Alexander duality on S we infer from Theorem the following formula of

Goresky MacPherson

Corollary Goresky MacPherson

n n

c

Let A be a nite set of ane subspaces in R Let M be the complement S U and

A A

let P be the intersection poset of A Then

M

i

e e

H P Z H M Z

p A

co dimpi

pP

where co dimp denotes the real codimension of the subspace corresponding to p

Analogous results for arrangements of spheres and pro jective spaces can b e found

in Goresky MacPherson and in In the following we describ e a simpler

more general and more p owerful approach that provides combinatorial formulas for quite

n

n n

general Grassmannian arrangements Let A b e a central arrangement in R or C H

with intersection p oset P and a dimension function d P N Let D D A fA g

p pP

b e the corresp onding P diagram of linear spaces In case each of the linear subspaces A

p

p P is invariant under the action of a nite or just closed subgroup G O n R

resp U n S pn of the orthogonal unitary symplectic group a natural step to make

is to dene the asso ciated orbit diagram More generally if T is an op eration a functor

n

asso ciating a space T V to a linear subspace V K where K R C or H then T A

denotes the diagram T A fT A g asso ciated to the corresp onding arrangement

p pP

n

of subspaces in T K There are several examples that come up very naturally in the

mathematical practice For example if V S V is the op eration of asso ciating the

n

unit sphere to the linear subspace V R then SA fSA g is the asso ciated

p pP

spherical diagram Similarly functors V R P V V C P V or simply P V in

b oth cases and in the case of quaternionic spaces lead to the corresp onding pro jective

arrangements R PA C PA or H P A denoted simply by P A Pro jective diagrams are

sp ecial cases of the asso ciated Grassmann diagrams obtained with the aid of the functor

V G V fL V dimV k g Lens space arrangements L A are dened

k m

similarly where L V L S V S V Z is the lens space asso ciated to the unit

m m m

sphere in a complex linear space V

We illustrate how the Homotopy Lemma can b e applied to pro duce a combi

natorial description of the link of the related arrangement in all the sp ecial cases ab ove

We obtain in particular simpler stronger and more natural pro ofs of some results from

Thms and ab out pro jective arrangements The advantage of the pro of

b elow is that it uses the Homotopy Lemma in its simplest form and allows a uniform

treatment of all the sp ecial cases ab ove It is clear that this metho d may b e useful for

other applications say for other group actions since the argument no longer requires the

i

arrangement to b e shifted to a more sp ecial p osition by a dilatation e e i n

i i

for some

Our ob jective is to show that the homotopy typ e of the link has a purely combinatorial

description in terms of the p oset P and the asso ciated rank function d P N The

link of a spherical ane pro jective lens Grassmann etc arrangement is the union of

all spaces in the arrangement It follows from the Projection Lemma that the link has

the same homotopy typ e as the homotopy colimit of the corresp onding diagram

We uniformly construct a combinatorially dened diagram which serves for comparison

n n

with the original one Cho ose a ag F fF g fg F F F K

i n

i

Let V T V b e one of the functors describ ed ab ove Then T F T F A the ag

diagram asso ciated with T A is dened by T F T F where the morphisms

p

dp

T F T F are the obvious inclusion maps Every two ag diagrams T F and T F

p q

are naturally isomorphic thus the isomorphism typ e of T F dep ends only on P and d

We want to compare our diagram T A with the combinatorially dened diagram

T F A There do es not seem to exist a natural map of diagrams b etween T A and

T F A b ecause eg the pro jection map is not natural This diculty was overcome in

in the case of pro jective diagrams by shifting the diagram T F A to a more sp ecial

p osition and by applying a more general version of the Homotopy Lemma by Vogt

that allows noncommutative diagrams if they commute up to coherent homotopies The

pro of of Theorem shows that in practical problems a very natural idea to use is the

comparison with the third so called ample space diagram that contains b oth T A and

T F A as sub diagrams We need a lemma which explains what is meant by ample in

all the interesting cases ab ove

n

Lemma Let T VectK Top V T V be one of the functors dened above

n

which to every vector space V K associates the corresponding projective space P V

n

Grassmann manifold G V the lens space L V for V C or the unit sphere

k m

n

S V Then in each of these cases there exists a functor R Vect K Top which

T

n n

associates with each subspace V K an ample subspace R V T K and to each

T

inclusion V W an inclusion of spaces R V R W so that the fol lowing condition

T T

is satised

n

Let V K dimV k Then for any W of dimension k with dimW V

there is an inclusion T W R V so that the inclusion map

T

i T W R V

W T

is a homotopy equivalence actual ly an inverse to a deformation retraction

Pro of The space R V is dened in a very similar way for all the examples ab ove For

T

n

example in the case of the functor V S V we put R V S K nS V The

T

construction in the case of a pro jective functor V P V is analogous one removes the

n n

pro jective space P V from P K The inclusion map i P V P K nP V is a

V

n

homotopy equivalence since P K nP V is the total space of a vector space bundle a

n

tubular neighb orho o d over P V Obviously P W P K nP V for any W with the

prop erty W V fg Finally i P W R V is a homotopy equivalence since

W T

n n

there exists a linear map O K K which maps V to W and leaves V invariant

V W

Something similar is done in the case of Grassmannians Here R V is dened by

G

k

n

R V fL G K dimL V g In this case R V is also the total

G k G

k k

space of a vector bundle over G V which can b e seen as follows If M V is the Stiefel

k k

k

manifold of all linear maps matrices D K V then G V M V Gk where

k k

n

Gk GLk K is the appropriate group of linear automorphisms Let p M K

k

n k

G K b e the pro jection map Note that p R V M V LK V where

k G k

k

k

LK V is the space of all linear maps and that the group Gk acts diagonally on the

k

pro duct M V LK V Now it is enough to recall that if X and Y are two Gspaces

k

and if the action on Y is free then the orbit space X Y G of the diagonal action is

represented as a b er bundle

X X Y G Y G

k

which in our case means that R V is b ered over G V with the b er LK V

G k

k

Hence i is a homotopy equivalence It is shown that i is also a homotopy equivalence

V W

analogously to the case of pro jective diagrams

n

V L C nL V The Finally in the case of the lens space functor let R

m m L

m

V is ample in the sense ab ove is analogous and can rely on the fact pro of that R

L

m

n

that the sphere S C is a join of spheres S V and S V and that the action of the

cyclic group Z resp ects this decomp osition

m

Theorem Homotopy types of Arrangements

n

Let A fA g be a linear subspace arrangement of K where K is one of the skew

p pP

elds R C or H Let P be the intersection poset of A with the dimension function

d P N dp dimA dened above and set P fp P dp k g for k

p k

Let T Vect Top V T V be the projective sphere Grassmann or the lens

space functor dened above and let T A be the corresponding arrangement of subspaces

n

of T K Then there is a homotopy equivalence

n

k

T K P T A

p

k

pP

k

i

Pro of We start by cho osing the ag F F span fe g in suciently general p osition

i k

k

with resp ect to the arrangement A This requirement means that for any F and A A

i p

if dimF dimA then A F fg Let us dene the asso ciated ample space

i p p

i

diagram R RA by R fR g R R A R F where V R V is

p pP p T p T dp T

the ample space functor asso ciated with T describ ed in Lemma Hence there exist

two naturally dened maps of diagrams and

T A RA T F A

induced by the inclusion maps T A R and T F R By Lemma

p p p p p p

these inclusion maps are homotopy equivalences From here and the Homotopy Lemma

it follows that ho colimT A ho colimR and ho colimT F A ho colimR which implies

ho colim T A ho colim T F A

n

Finally we notice that the emb eddings T F T K induce a diagram map

k

T F A F

n

where F denotes the constant P diagram which has F T K for all p P and

p

b

0

for p p From the Embedding Lemma we get that is an identity maps f

pp

emb edding

n

b

T K P ho colim T F A ho colimF

S

n

k

b

and we easily identify the image of with the space T K P

k

k

Note that for the homotopy formula for Grassmann arrangements in Theorem

only the truncated p oset P fp P dp k g is relevant This corresp onds

k

to the fact that spaces with dp k do not have k dimensional subspaces so A and

A fA A dp k g have the same asso ciated k Grassmann arrangement

p

k

The following prop osition shows that there is a general decomp osition formula for the

homology of ag diagrams and a p osteriori of the T links of K arrangements for K R

C or H

Prop osition Homology of Flag Diagrams

n

Let A be an arrangement of linear subspaces in K and let T be one of the functors

described above Let T F A be the combinatorial ly dened ag diagram associated to the

arrangement A and the functor T We set s minfdA g and t maxfdA g

p pP p pP

Assume that for the coecient ring R and for al l s k m t

m k

the exact sequence of the pair T K T K splits in homology and

m

the homology groups H T K R are free R modules

Then H ho colimT F A R

t

M

k k

s

H P R H T K T K R H P R H T K R

k s

k s

Pro of For this pro of we x the co ecient ring R used for homology computations and

we write H for H R For an arbitrary p oset Q and a natural numb er k we intro duce

k

constant diagrams M over Q dened by M Q T op M q T K If it

k Q k Q k Q

is clear from the context which p oset Q is used we write M for the diagram M Note

k k Q

that our diagram T F A can b e squeezed in b etween two constant diagrams M and

sP

k

M Clearly ho colimM Q T K The Kunneth theorem and the freeness of

tP k Q

k

R mo dules H T K imply

s

H Q H T K H ho colimM

sQ

k m

From this observation and from the fact that the map H T K H T K is a

monomorphism for k m we conclude that the map ho colimM ho colimM for

k m

k m induces a monomorphism of homology groups Then the comp osition of homo

morphisms

H ho colimM H ho colimT F A H ho colimM

s t

is a monomorphism to o Thus the rst of them is also a monomorphism It follows that

the long exact sequence of the pair ho colimT F A ho colimM splits and

s

H ho colimM H ho colimT F A ho colimM H ho colimT F A

s s

Informally sp eaking we p eel from the homotopy colimit of the diagram T F A the part

of its homology coming from the constant sub diagram M Let E b e the restriction

s i

of the diagram T F A on the p oset P i By excision recalling the denition

si

of the homotopy colimit ho colimT F A we have H ho colimT F A ho colimM

s

By induction on i we may assume that for some i H ho colimE ho colimM

sP

[s+1]

s

H P H T K H ho colimT F A

s

si

M

k k

H P H T K T K

k

k s

H ho colimE ho colimM

siP

i

[s+i+1]

The long exact sequence of the triple

ho colimE ho colimM ho colimM

siP siP

i

[s+i+1] [s+i+1]

splits by the same argument as ab ove This means that we can p eel from the homology

H ho colimE ho colimM the part isomorphic to

siP

i

[s+i+1]

H ho colimM ho colimM

siP siP

[s+i+1] [s+i+1]

si si

H P H T K T K

si

The part that remains is isomorphic to the group H ho colimE M The

i siP

[s+i+1]

last group is by excision isomorphic to H ho colimE ho colimM So the

siP

i

[s+i+2]

Kunneth formula the pro cess describ ed ab ove and induction on i lead to the desired

formula

If the arrangement A is essential ie if s and t maxfdA g n then b oth

p pP

K and T K are interpreted as empty spaces and the formula given in Prop osition

can b e rewritten as follows

Corollary Let A be an essential arrangement satisfying the assumptions of Propo

sition Then

n

M

k k

H P R H T K T K R H ho colimT F A R

k

k

For example if T P is the functor which asso ciates the complex pro jective space

P V to every complex linear space V then the formula given in Corollary has the

following form

n

M

H P R H ho colimT F A R

r k r

k

k

where r and H X R for all k This formula together with its counterpart

k

for the real pro jective arrangements was formulated and proved in Unfortunately

as it was kindly p ointed to us by Anders Bjorner and Karanbir Sarkaria the formulation

there suers from some misprints

The following example which we also owe to A Bjorner and K Sarkaria shows how a

formula of the typ e ab ove arises in connection with the StanleyReisner ring of a simplicial

complex

Corollary Let be a simplicial complex on the vertex set f ng Let A be

n

the arrangement of complex linear subspaces in C dened by A fA g where

n

A span fe g e the ith unit coordinate vector in C Let P A fP A g

j j i

C

S

be the associated projective arrangement Then the homology of the union P A of

the arrangement P A is given by

n

M

H Z H P A Z

r k r

k

k

where f dim k g is a subposet of and its order complex

k k

Note that the union of the arrangement P A is a pro jective variety whose homoge

neous co ordinate ring is the StanleyReisner ring of

i i

Note that Prop osition deals with the case when H T K H T K is

injective in contrast to the GoreskyMacPherson formula Corollary We already

mentioned that the GoreskyMacPherson formula for the cohomology of the complement

of an arrangement A of ane subspaces can b e proved by Alexander duality from the

homology of an asso ciated arrangement S A of spheres In the case when T S the map

i i

e e

H T K H T K is trivial Although it do es not completely t in the setting of

this section one may regard toric varieties seen from the p oint of view of Section

as an interesting third case when the map in homology induced by the diagram maps

are surjective

Subgroup Complexes

i

The order complex of the p oset S G fP G jP j p g of nontrivial p

p

subgroups of a nite group G has received considerable interest over that past few years

see for example It was already observed by Quillen that S G is homotopy

p

equivalent to the p oset A G of nontrivial elementary ab elian psubgroups of G In

p

the authors consider the covering of A G by the sub complexes A NA for a

p p

xed solvable normal p subgroup N and maximal elementary ab elian psubgroups A of

G Then they use the following facts

a Intersections of the spaces of typ e A NA are again of the typ e A ND

p p

for some elementary ab elian psubgroup D of G

b For a solvable normal p group N and an elementary ab elian psubgroup A the

complex A NA is homotopy equivalent to a wedge of spheres of dimension

p

rankD

Observation a follows by basic group theoretical argumentation Assertion b is

much less obvious It was established by Quillen Theorem but also follows by

applications of the homotopy colimit metho ds see Theorem A Using facts a

and b the Projection the Homotopy and the Wedge Lemma the following wedge

decomp osition of A G for nite solvable groups G with nontrivial normal p group

p

is proved in

Theorem Pulkus Welker Theorem B

Let G be a nite group and let p be a prime Let N be a solvable normal p subgroup Let

C N N be the intersection of al l maximal elementary abelian psubgroups of GN For

AN N A GN let c be an arbitrary but xed point in A GN Then

p AN N p AN N

S G is homotopy equivalent to

p

A NA A GN

p p AN N

AN N A GN fC N N g

p

C N N

where the wedge is formed by identifying for AN N N N

the point c A GN A NA

AN N p AN N p

with the point AN N A GN A N

p p

In particular if A is a maximal elementary abelian group of rank r in G then

A NA A GN is homotopic to a wedge of r spheres

p p

AN N

Acknowledgments

Thanks to EvaMaria Feichtner for helpful comments on the manuscript We are grate

ful to Emmanuel Dror Farjoun and Rainer Vogt for some corrections and very valuable

remarks and references The referees remarks on the pap er were extremely helpful and

hop efully led to a substantial improvement of the exp osition

We all thank the KonradZuseZentrum in Berlin for its hospitality

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