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Home , Triangulation (topology)

ISSN online printed

Algebraic Geometric

Volume

ATG

Published Decemb er

Equivalences to the triangulation

Duane Randall

Abstract We utilize the obstruction theory of GalewskiMatumotoStern

to derive equivalent formulations of the Triangulation Conjecture For ex

n

ample every closed top ological M with n can b e simplicially

triangulated if and only if the two distinct combinatorial triangulations of

5

RP are simplicially concordant

AMS Classication N S Q

Keywords Triangulation KirbySieb enmann class Bo ckstein op erator

top ological manifold

Intro duction

The Triangulation Conjecture TC arms that every closed top ological man

n

ifold M of dimension n admits a simplicial triangulation The vanishing

of the KirbySieb enmann class KS M in H M Z is b oth necessary and

n

sucient for the existence of a combinatorial triangulation of M for n

n

by A combinatorial triangulation of a closed manifold M is a simplicial

triangulation for which the link of every i is a combinatorial of

dimension n i Galewski and Stern Theorem and Matumoto

n

indep endently proved that a closed connected top ological manifold M with

n is simplicially triangulable if and only if

KS M in H M ker

where denotes the Bo ckstein op erator asso ciated to the exact sequence

ker Z of ab elian groups Moreover the Triangulation

Conjecture is true if and only if this exact sequence splits by or page

The Ro chlin invariant morphism is dened on the b ordism

group of oriented homology mo dulo those which b ound acyclic

compact PL Fintushel and Stern and Furuta proved that

is innitely generated

We freely employ the notation and information given in Ranickis excellent ex

p osition The relative b oundary version of the GalewskiMatumotoStern

c

Geometry Topology Publications

Duane Randall

obstruction theory in pro duces the following result Given any homeo

morphism f jK j jLj of the p olyhedra of closed m dimensional PL man

ifolds K and L with m f is homotopic to a PL if

and only if KS f vanishes in H L Z More generally a homeomorphism

f jK j jLj is homotopic to a PL map F K L with acyclic p oint inverses

if and only if

KS f in H L ker

n

Concordance classes of simplicial triangulations on M for n corresp ond

bijectively to vertical homotopy classes of liftings of the stable top ological tan

gent bundle M BTOP to BH by Theorem and so are enumerated

by H M ker The classifying space BH for the stable bundle theory as

so ciated to combinatorial homology manifolds in is denoted by BTRI in

and by BHML in We employ obstruction theory to derive some known

and new results and generalizations of and on the existence of simplicial

triangulations in section and to record some equivalent formulations of TC

in section Although some of these formulations may b e known they do not

seem to b e do cumented in the literature

Simplicial Triangulations



Let denote the integral Bo ckstein op erator asso ciated to the exact sequence



Z Z Z We pro ceed to derive some consequences of



the vanishing of on KirbySieb enmann classes The co ecient group for

is understo o d to b e Z whenever omitted Matumoto knew in



that the vanishing of KS M implied the vanishing of KS M Let

m

denote the fundamental class of the Eilenb ergMacLane space K Z m Since

m

H K Z m G for all co ecient groups G trivially in

m

m

H K Z m ker Thus vanishes on KS M in or KS f in



whenever do es This observation together with and justies the

following wellknown statements Every closed connected top ological manifold

n 

M with n and KS M admits a simplicial triangulation Let

f jK j jLj b e any homeomorphism of the p olyhedra of closed m dimensional



PL manifolds K and L with m If KS f then f is homotopic to

a PL map F K L with acyclic p oint inverses

Prop osition All k fold Cartesian pro ducts of closed manifolds are sim

plicially triangulable for k All pro ducts M S with nonorientable closed

Algebraic Geometric Topology Volume

Equivalences to the triangulation conjecture

manifolds M are simplicially triangulable Let N b e any simply connected

closed manifold with KS N trivial and also b rank of H N Z Let

f jK j jLj b e any homeomorphism with KS f nontrivial and jK j jLj

N S Then f is homotopic to a PL map F K L with acyclic p oint

inverses

Pro of of Since KS is a primitive cohomology class for the universal

bundle on BTOP we have KS M M KS M KS M in

H M M Triviality of on H M by dimensionality yields triangu

lability of all k fold pro ducts of closed manifolds for k and of M S

by

b

The pro duct N S admits distinct combinatorial structures by more

over for every nonzero class u in H N S there is a homeomorphism of

p olyhedra with distinct combinatorial structures whose CassonSullivan invari



ant is u by page The vanishing of KS f follows from the triviality



of on H N S H N Z H S Z

No closed manifold M with KS M nonzero can b e simplicially triangu

lated Yet k fold pro ducts of such manifolds M by and their pro ducts

with spheres or tori pro duce innitely many distinct noncombinatorial yet

simplicially triangulable closed manifolds in every dimension In contrast

there are no known examples of nonsmo othable closed manifolds which can

b e simplicially triangulated according to Problem of page

n

Theorem Let M b e any closed connected top ological manifold with

n such that the stable spherical bration determined by the tangent bundle

M has o dd order in M B S G Supp ose that either H M Z has no

torsion or else all torsion in H M Z has order Then M is simplicially

triangulable

Pro of The StiefelWhitney classes of M are trivial by the hyp othesis of o dd

order We rst consider the sp ecial case that M is stably b er homotopically

trivial Let g M S GSTOP b e any lifting of a classifying map M M

BSTOP in the bration

j

S GSTOP BSTOP BSG



e

The Postnikov stage of S GSTOP is K Z K Z Now j KS

e

by Theorem of page where denotes the universal

  

e

bundle over BSTOP Clearly u where u generates j KS

H K Z Z Z If all nonzero torsion in H M Z has order

Algebraic Geometric Topology Volume

Duane Randall

   

then KS M g u If H M Z has no torsion then g



so again KS M Thus KS M

We supp ose now that the stable spherical bration of M has order a

in M B S G with a Let s M S a M b e a section to the sphere

bundle pro jection p S a M M asso ciated to a M Now S a M

is a stably b er homotopically trivial manifold since its stable tangent bundle

 

is a p M Since KS M a KS M s KS S a M

we conclude that

   

KS M s KS S a M s

We consider the following homotopy commutative diagram of principal bra

tions

i

K ker K ker K ker

i

y y y

i t

BH B H B P L K

y y y

c

k s i KS

S BTOP B T O P B P L K Z

c

y y

KS

K ker K ker

The b er map is induced from the pathlo op bration on K ker via the

Bo ckstein op erator on the fundamental class of K Z The induced

morphism on is the Ro chlin morphism Z by construction



d

The relative principal bration is induced from via the map KS classifying



d

the relative universal KirbySieb enmann class Thus KS i KS

Inclusion maps are denoted by i in The induced morphisms t and



d

KS are isomorphisms on We employ in the pro of of Theorem



Equivalent formulations to TC

Galewski and Stern constructed a nonorientable closed connected manifold

M in such that S q KS M generates H M Z They also proved

that any such M is universal for TC Moreover Theorem of es

sentially arms that either TC is true or else no closed connected top ological

n

n manifold M with S q KS M and n can b e simplicially triangu

lated

Algebraic Geometric Topology Volume

Equivalences to the triangulation conjecture

Theorem

The following statements are equivalent to the Triangulation Conjecture

d

Any equivalently all of the classes KS KS and in is

trivial if and only if any equivalently all of the b er maps and

in admits a section

The essential map f S e BTOP lifts to BH in



S q KS in H BH for the universal bundle on BH

n

Any closed connected top ological manifold M with S q KS M and

n admits a simplicial triangulation

Every homeomorphism f jK j jLj with KS f nontrivial is homo

topic to a PL map with acyclic p oint inverses where K and L are any

combinatorially distinct p olyhedra with jK j jLj N RP Here

N denotes any simply connected closed manifold with KS N trivial

and p ositive rank for H N Z

All combinatorial triangulations of each closed connected PL manifold

n

M with n are concordant as simplicial triangulations

The two distinct combinatorial triangulations of RP are simplicially

concordant

n

Every closed connected top ological manifold M with n that is

stably b er homotopically trivial admits a simplicial triangulation

Pro of TC Statement is equivalent to the splitting of the exact

sequence ker Z through the induced morphisms on

homotopy in dimension

TC Let k s S BTOP represent the KirbySieb enmann class in

homotopy That is k s has order and is dual to KS under the mo d

Hurewicz morphism Now k s admits an extension f S e BTOP since

the cobration exact sequence



BTOP S e BTOP BTOP BTOP



corresp onds to Z Z Z Z Z If g S e BH is

any lifting of f the comp osite map using

g

i t

h S S e BH B H B P L K

pro duces u h in with u and u since u h

d

KS k s generates K Z Thus TC is true Conversely if TC is

true a section s BTOP BH to in gives a lifting s f of f

Algebraic Geometric Topology Volume

Duane Randall

TC Prop erties of KS are enumerated in and Since

S q KS a section s to in gives S q KS so TC im

plies We now assume that TC is false and claim that the generator S q for

H K Z Z lies in the image of

H K ker H om K ker Z H omker Z



The Serre exact sequence then gives S q in H K so

 

S q KS t i S q

Thus we must construct a morphism ker Z which do es not extend to



We consider the sequence ker ker ker Z and dene

h ker Z Z as follows hv if and only if v z for some

z with z Now h is a welldened and nontrivial morphism since

do es not have an element u with u and u by hyp othesis The

comp osite morphism h ker Z do es not extend to

n

TC Supp ose M with S q KS M admits a simplicial triangu



lation Now S q KS M g S q KS for any lifting g M BH of

M BTOP Since S q KS TC holds by

d

TC Clearly triviality of KS in gives KS f via naturality

for every f Supp ose that KS f for any such f in Now KS f

  1

v i a in H M Z H RP H L Here a generates H RP

1 1 1

and i RP RP Naturality via the universal example CP RP

   1

for v i a gives KS f v i a Since i H RP ker



H RP ker is a monomorphism i a if and only if a Now

a if and only if TC is true via the bration

1

K ker K RP

TC TC holds if and only if for the fundamental class

n

of K Z Concordance classes of simplicial triangulations of M arising

from combinatorial triangulations dier by classes in H M This subgroup

of H M ker is trivial by naturality if Conversely H RP

if the two distinct combinatorial triangulations of RP given by Theorem

in pages and are simplicially concordant But a if and

K Z and naturality for only if via the skeletal inclusion RP

RP RP

TC Similar to Theorem of we consider a regular neighb orho o d

m

of the skeleton of S GSTOP emb edded in R for some m in order

Algebraic Geometric Topology Volume

Equivalences to the triangulation conjecture

to obtain a smo othly parallelizable manifold W with b oundary and a map

g W S GSTOP which is a homotopy equivalence through dimension

b

The double DW is smo othly parallelizable and admits an extension g DW



b

S GSTOP Note that g is a monomorphism through dimension Let h

M DW b e a degree one normal map Now M is stably b er homotopically

 

b

trivial and h is a monomorphism in cohomology In particular g h is

a monomorphism on H S GSTOP ker We conclude that KS M



b

g h if and only if for the fundamental class of

K Z So statement yields

Let f K Z K Z classify Since assuming statement

f admits a lifting h K Z K in such that f h

The diagram

CP K

h





y y

f



CP K Z Z Z CP K Z

yields a splitting to the exact sequence ker Z so TC

holds

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Duane Randall

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Department of Mathematics and Computer Science

Loyola University New Orleans LA USA

Email randallloynoedu Received July

Algebraic Geometric Topology Volume