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Torus Actions and Their Applications in Topology and Combinatorics Torus actions and their applications in top ology and combinatorics Victor M Buchstab er Taras E Panov Author address Department of Mathematics and Mechanics Moscow State Univer sity Moscow RUSSIA Email address buchstabmechmathmsusu Department of Mathematics and Mechanics Moscow State Univer sity Moscow RUSSIA Email address tpanovmechmat hmsusu Mathematics Subject Classication B Q R M B F C Abstract Here the study of torus actions on top ological spaces is presented connecting combinatorial and convex geometry with commutative and as a bridge homological algebra algebraic geometry and top ology This link helps in under standing the geometry and top ology of a space with torus action by studying the combinatorics of the space of orbits Conversely the most subtle prop erties of a combinatorial ob ject can b e recovered by realizing it as the orbit structure for a prop er manifold or complex acted on by a torus The latter can b e a symplectic manifold with Hamiltonian torus action a toric variety or manifold a subspace arrangement complement etc while the combinatorial ob jects include simplicial and cubical complexes p olytop es and arrangements This approach also provides a natural top ological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics The exp osition centers around the theory of momentangle complexes provid ing an eectiv e way to study triangulations by metho ds of equivariant top ology The b o ok includes many new and wellknown op en problems and would b e suitable e hop e that it will b e useful for sp ecialists b oth in top ology and as a textb o ok W in combinatorics and will help to establish even tighter connections b etween the sub jects involved Contents Intro duction olytop es Chapter P Denitions and main constructions Face vectors and DehnSommerville equations theorem The g Upp er Bound and Lower Bound theorems StanleyReisner face rings of simple p olytop es Chapter Top ology and combinatorics of simplicial complexes Abstract simplicial complexes and p olyhedrons Basic P L top ology and op erations with simplicial complexes Simplicial spheres Triangulated manifolds Bistellar moves Chapter Commutative and homological algebra of simplicial complexes StanleyReisner face rings of simplicial complexes CohenMacaulay rings and complexes Homological algebra background Homological prop erties of face rings Toralgebras and Betti numb ers Gorenstein complexes and DehnSommerville equations Chapter Cubical complexes Denitions and cubical maps Cubical sub divisions of simple p olytop es and simplicial complexes Chapter Toric and quasitoric manifolds Toric varieties Quasitoric manifolds Stably complex structures and quasitoric representatives in cob ordism classes Combinatorial formulae for Hirzebruch genera of quasitoric manifolds Classication problems Chapter Momentangle complexes Momentangle manifolds Z dened by simple p olytop es P General momentangle complexes Z K Cell decomp ositions of momentangle complexes vii viii CONTENTS Momentangle complexes corresp onding to joins connected sums and bistellar moves Borel constructions and DavisJan uszkiewicz space Walk around the construction of Z generalizations analogues and K additional comments Chapter Cohomology of momentangle complexes and combinatorics of triangulated manifolds b ergMo ore sp ectral sequence The Eilen Cohomology algebra of Z K Bigraded Betti numb ers of Z the case of general K K Bigraded Betti numb ers of Z the case of spherical K K Partial quotients of Z P Bigraded Poincare duality and DehnSommerville equations Chapter Cohomology rings of subspace arrangement complements General arrangements and their complements Co ordinate subspace arrangements and the cohomology of Z K Diagonal subspace arrangements and the cohomology of Z K Bibliography Index Intro duction Torus actions on top ological spaces is classical and one of the most develop ed elds in equivariant top ology Sp ecic problems connected with torus actions arise in dif ferent areas of mathematics and mathematical physics which results in p ermanent interest in the theory constant source of new applications and p enetration of new ideas in top ology Many volumes devoted to particular asp ects of this wide eld of mathematical knowledge are available The top ological approach is the sub ject of monograph by G Bredon Monograph by M Audin deals with torus actions from the symplectic geometry viewp oint The algebrogeometrical part of geometry is the study known as the geometry of toric varieties or simply toric presented in several texts These include V Danilovs original survey article and more recent monographs by T Oda W Fulton and G Ewald The orbit space of a torus action carries a rich combinatorial structure In many cases studying the combinatorics of the quotient is the easiest and the most ecient way to understand the top ology of a toric space This approach works in the op p osite direction as well the equivariant top ology of a torus action sometimes helps to interpret and prove the most subtle combinatorial results top ologically In the most symmetric and regular cases such as pro jective toric varieties or Hamiltonian torus actions on symplectic manifolds the quotient can b e identied with a convex p olytop e More general toric spaces give rise to other combinatorial structures re lated with their quotients Examples here include simplicial spheres triangulated manifolds general simplicial complexes cubical complexes subspace arrangements etc Combined applications of combinatorial top ological and algebrogeometrical metho ds stimulated intense development of toric geometry during the last three decades This remarkable conuence of ideas enriched all the sub jects involved with a numb er of sp ectacular results Another source of applications of top ological and algebraic metho ds in combinatorics was provided by the theory of Stanley Reisner face rings and CohenMacaulay complexes describ ed in R Stanleys mono Our motivation was to broaden the existing bridge b etween torus graph actions and combinatorics by giving some new constructions of toric spaces which naturally arise from combinatorial considerations We also interpret many exist ing results in such a way that their relationships with combinatorics b ecome more transparent Traditionally simplicial complexes or triangulations were used in top ology as a to ol for combinatorial treatment of top ological invariants of spaces or manifolds On the other hand triangulations themselves can b e regarded as particular structures so the space of triangulations b ecomes the ob ject of study The idea of considering the space of triangulations of a given manifold has b een INTRODUCTION also motivated by some physical problems One gets an eective way of treat ment of combinatorial results and problems concerning the numb er of faces in a triangulation by interpreting them as extremal value problems on the space of tri angulations We implement some of these ideas in our b o ok as well by constructing and investigating invariants of triangulations using the equivariant top ology of toric spaces The b o ok is intended to b e a systematic but elementary overview for the asp ects of torus actions mostly related to combinatorics However our level of exp osition is not balanced b etween top ology and combinatorics We do not assume any par ticular readers knowledge in combinatorics but in top ology a basic knowledge of characteristic classes and sp ectral sequences techniques may b e very helpful in the last chapters All necessary information is contained for instance in S No vikovs b o ok We would recommend this b o ok since it is reasonably concise has a rather broad scop e and pays much attention to the combinatorial asp ects of top ology Nevertheless we tried to provide necessary background material in the algebraic top ology and hop e that our b o ok will b e of interest to combinatorialists as well A signicant part of the text is devoted to the theory of momentangle com plexes currently b eing develop ed by the authors This study was inspired by pap er of M Davis and T Januszkiewicz where a top ological analogue of toric varieties was intro duced In their work Davis and Januszkiewicz used a certain m space Z assigned to every simplicial complex on the vertex set universal T K m f mg In its turn the denition of Z was motivated see x by K the construction of the Coxeter complex of a Coxeter group and its generalizations by E Vinb erg Our approach brings the space Z to the center of attention To each subset K m k mk m there is assigned a canonical T equivariant emb edding D T m m m D where D is the standard p olydisc in C and k is the cardinality of This corresp ondence extends to any simplicial complex K on m and pro duces a m canonical bi graded cell decomp osition of the DavisJanuszkiewicz T space Z K which we refer to as the momentangle complex There is also a more general version of momentangle complexes dened for any cubical sub complex in a unit cub e see section The construction of Z gives rise to a functor see Prop osition K m from the category of simplicial complexes and inclusions to the category of T spaces and equivariant maps This functor induces a homomorphism b etween the standard simplicial chain complex of a simplicial pair K K and the bigraded cellular chain complex of Z Z The remarkable prop erty of the functor
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