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Equivalences to the Triangulation Conjecture 1 Introduction ISSN online printed Algebraic Geometric Topology Volume ATG Published Decemb er Equivalences to the triangulation conjecture 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 manifold 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 simplex is a combinatorial sphere 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 homology b ordism group of oriented homology spheres mo dulo those which b ound acyclic compact PL manifolds 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 homeomorphism 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 cohomology 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
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