Mathematical Surveys

and Monographs

Volume 204

American Mathematical Society

Toric Topology

Victor M. Buchstaber Taras E. Panov

Toric Topology

Mathematical Surveys

and Monographs

Volume 204

Toric Topology

Victor M. Buchstaber Taras E. Panov

American Mathematical SocietyProvidence, Rhode Island

http://dx.doi.org/10.1090/surv/204

EDITORIAL COMMITTEE

Robert GuralnickMichael A. Singer, ChairBenjamin Sudakov

Constantin TelemanMichael I. Weinstein

2010 Mathematics Subject Classification. Primary 13F55, 14M25, 32Q55, 52B05, 53D12,55N22, 55N91, 55Q15 57R85, 57R91.

For additional information and updates on this book, visitwww.ams.org/bookpages/surv-204

Library of Congress Cataloging-in-Publication Data

Buchstaber, V. M.Toric topology / Victor M. Buchstaber, Taras E. Panov.

pages cm. – (Mathematical surveys and monographs ; volume 204)Includes bibliographical references and index.ISBN 978-1-4704-2214-1 (alk. paper)1. Toric varieties. 2. Algebraic varieties. 3. Algebraic topology. 4. Geometry, Algebraic.

I. Panov, Taras E., 1975– II. Title

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Contents

Introduction ixChapter guide xAcknowledgements xiii

Chapter 1. Geometry and Combinatorics of Polytopes 11.1. Convex polytopes 11.2. Gale duality and Gale diagrams 91.3. Face vectors and Dehn–Sommerville relations 141.4. Characterising the face vectors of polytopes 19Polytopes: Additional Topics 251.5. Nestohedra and graph-associahedra 251.6. Flagtopes and truncated cubes 381.7. Differential algebra of combinatorial polytopes 441.8. Families of polytopes and differential equations 48

Chapter 2. Combinatorial Structures 552.1. Polyhedral fans 562.2. Simplicial complexes 592.3. Barycentric subdivision and flag complexes 642.4. Alexander duality 662.5. Classes of triangulated spheres 692.6. Triangulated manifolds 752.7. Stellar subdivisions and bistellar moves 782.8. Simplicial posets and simplicial cell complexes 812.9. Cubical complexes 83

Chapter 3. Combinatorial Algebra of Face Rings 913.1. Face rings of simplicial complexes 923.2. Tor-algebras and Betti numbers 973.3. Cohen–Macaulay complexes 1043.4. Gorenstein complexes and Dehn–Sommerville relations 1113.5. Face rings of simplicial posets 114Face Rings: Additional Topics 1213.6. Cohen–Macaulay simplicial posets 1213.7. Gorenstein simplicial posets 1253.8. Generalised Dehn–Sommerville relations 127

Chapter 4. Moment-Angle Complexes 1294.1. Basic definitions 1314.2. Polyhedral products 135

v

vi CONTENTS

4.3. Homotopical properties 1394.4. Cell decomposition 1434.5. Cohomology ring 1444.6. Bigraded Betti numbers 1504.7. Coordinate subspace arrangements 157Moment-Angle Complexes: Additional Topics 1624.8. Free and almost free torus actions on moment-angle complexes 1624.9. Massey products in the cohomology of moment-angle complexes 1684.10. Moment-angle complexes from simplicial posets 171

Chapter 5. Toric Varieties and Manifolds 1795.1. Classical construction from rational fans 1795.2. Projective toric varieties and polytopes 1835.3. Cohomology of toric manifolds 1855.4. Algebraic quotient construction 1885.5. Hamiltonian actions and symplectic reduction 195

Chapter 6. Geometric Structures on Moment-Angle Manifolds 2016.1. Intersections of quadrics 2016.2. Moment-angle manifolds from polytopes 2056.3. Symplectic reduction and moment maps revisited 2096.4. Complex structures on intersections of quadrics 2126.5. Moment-angle manifolds from simplicial fans 2156.6. Complex structures on moment-angle manifolds 2196.7. Holomorphic principal bundles and Dolbeault cohomology 2246.8. Hamiltonian-minimal Lagrangian submanifolds 231

Chapter 7. Half-Dimensional Torus Actions 2397.1. Locally standard actions and manifolds with corners 2407.2. Toric manifolds and their quotients 2427.3. Quasitoric manifolds 2437.4. Locally standard T -manifolds and torus manifolds 2577.5. Topological toric manifolds 2787.6. Relationship between different classes of T -manifolds 2827.7. Bounded flag manifolds 2847.8. Bott towers 2877.9. Weight graphs 302

Chapter 8. Homotopy Theory of Polyhedral Products 3138.1. Rational homotopy theory of polyhedral products 3148.2. Wedges of spheres and connected sums of sphere products 3218.3. Stable decompositions of polyhedral products 3268.4. Loop spaces, Whitehead and Samelson products 3318.5. The case of flag complexes 340

Chapter 9. Torus Actions and Complex Cobordism 3479.1. Toric and quasitoric representatives in complex bordism classes 3479.2. The universal toric genus 3579.3. Equivariant genera, rigidity and fibre multiplicativity 3649.4. Isolated fixed points: localisation formulae 367

CONTENTS vii

9.5. Quasitoric manifolds and genera 3769.6. Genera for homogeneous spaces of compact Lie groups 3809.7. Rigid genera and functional equations 385

Appendix A. Commutative and Homological Algebra 395A.1. Algebras and modules 395A.2. Homological theory of graded rings and modules 398A.3. Regular sequences and Cohen–Macaulay algebras 405A.4. Formality and Massey products 409

Appendix B. Algebraic Topology 413B.1. Homotopy and homology 413B.2. Elements of rational homotopy theory 424B.3. Eilenberg–Moore spectral sequences 426B.4. Group actions and equivariant topology 428B.5. Stably complex structures 433B.6. Weights and signs of torus actions 434

Appendix C. Categorical Constructions 439C.1. Diagrams and model categories 439C.2. Algebraic model categories 444C.3. Homotopy limits and colimits 450

Appendix D. Bordism and Cobordism 453D.1. Bordism of manifolds 453D.2. Thom spaces and cobordism functors 454D.3. Oriented and complex bordism 457D.4. Characteristic classes and numbers 463D.5. Structure results 467D.6. Ring generators 468

Appendix E. Formal Group Laws and Hirzebruch Genera 473E.1. Elements of the theory of formal group laws 473E.2. Formal group law of geometric cobordisms 477E.3. Hirzebruch genera (complex case) 479E.4. Hirzebruch genera (oriented case) 486E.5. Krichever genus 488

Bibliography 495

Index 511

Introduction

Traditionally, the study of torus actions on topological spaces has been consid-ered as a classical field of algebraic topology. Specific problems connected with torusactions arise in different areas of mathematics and mathematical physics, which re-sults in permanent interest in the theory, new applications and penetration of newideas into topology.

Since the 1970s, algebraic and symplectic viewpoints on torus actions haveenriched the subject with new combinatorial ideas and methods, largely based onthe convex-geometric concept of polytopes.

The study of algebraic torus actions on algebraic varieties has quickly devel-oped into a highly successful branch of algebraic geometry, known as toric geometry.It gives a bijection between, on the one hand, toric varieties, which are complexalgebraic varieties equipped with an action of an algebraic torus with a dense or-bit, and on the other hand, fans, which are combinatorial objects. The fan allowsone to completely translate various algebraic-geometric notions into combinatorics.Projective toric varieties correspond to fans which arise from convex polytopes. Avaluable aspect of this theory is that it provides many explicit examples of alge-braic varieties, leading to applications in deep subjects such as singularity theoryand mirror symmetry.

In symplectic geometry, since the early 1980s there has been much activity inthe field of Hamiltonian group actions on symplectic manifolds. Such an actiondefines the moment map from the manifold to a Euclidean space (more precisely,the dual Lie algebra of the torus) whose image is a convex polytope. If the torus hashalf the dimension of the manifold, the image of the moment map determines themanifold up to equivariant symplectomorphism. The class of polytopes which ariseas the images of moment maps can be described explicitly, together with an effectiveprocedure for recovering a symplectic manifold from such a polytope. In symplecticgeometry, as in algebraic geometry, one translates various geometric constructionsinto the language of convex polytopes and combinatorics.

There is a tight relationship between the algebraic and the symplectic pictures:a projective embedding of a toric manifold determines a symplectic form and amoment map. The image of the moment map is a convex polytope that is dual tothe fan. In both the smooth algebraic-geometric and the symplectic situations, thecompact torus action is locally isomorphic to the standard action of (S1)n on Cn

by rotation of the coordinates. Thus the quotient of the manifold by this actionis naturally a manifold with corners, stratified according to the dimension of thestabilisers, and each stratum can be equipped with data that encodes the isotropytorus action along that stratum. Not only does this structure of the quotient providea powerful means of investigating the action, but some of its subtler combinatorialproperties may also be illuminated by a careful study of the equivariant topology

ix

x INTRODUCTION

of the manifold. Thus, it should come as no surprise that since the beginning of the1990s, the ideas and methodology of toric varieties and Hamiltonian torus actionshave started penetrating back into algebraic topology.

By 2000, several constructions of topological analogues of toric varieties andsymplectic toric manifolds had appeared in the literature, together with differentseemingly unrelated realisations of what later has become known as moment-anglemanifolds. We tried to systematise both known and emerging links between torusactions and combinatorics in our 2000 paper [67] in Russian Mathematical Sur-veys, where the terms ‘moment-angle manifold’ and ‘moment-angle complex’ firstappeared. Two years later it grew into a book Torus Actions and Their Applicationsin Topology and Combinatorics [68] published by the AMS in 2002 (the extendedRussian edition [69] appeared in 2004). The title ‘Toric Topology’ coined by ourcolleague Nigel Ray became official after the 2006 Osaka conference under the samename. Its proceedings volume [177] contained many important contributions to thesubject, as well as the introductory survey An Invitation to Toric Topology: Ver-tex Four of a Remarkable Tetrahedron by Buchstaber and Ray. The vertices of the‘toric tetrahedron’ are topology, combinatorics, algebraic and symplectic geometry,and they have symbolised many strong links between these subjects. With manyyoung researchers entering the subject and conferences held around the world everyyear, toric topology has definitely grown into a mature area. Its various aspects arepresented in this monograph, with an intention to consolidate the foundations andstimulate further applications.

Chapter guide

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Each chapter and most sections have their own introductions with more specificinformation about the contents. ‘Additional topics’ of Chapters 1, 3 and 4 containmore specific material which is not used in an essential way in the following chapters.The appendices at the end of the book contain material of more general nature,

CHAPTER GUIDE xi

not exclusively related to toric topology. A more experienced reader may refer tothem only for notation and terminology.

At the heart of toric topology lies a class of torus actions whose orbit spacesare highly structured in combinatorial terms, that is, have lots of orbit types tiedtogether in a nice combinatorial way. We use the generic terms toric space and toricobject to refer to a topological space with a nice torus action, or to a space producedfrom a torus action via different standard topological or categorical constructions.Examples of toric spaces include toric varieties, toric and quasitoric manifolds andtheir generalisations, moment-angle manifolds, moment-angle complexes and theirBorel constructions, polyhedral products, complements of coordinate subspace ar-rangements, intersections of real or Hermitian quadrics, etc.

In Chapter 1 we collect background material related to convex polytopes, in-cluding basic convex-geometric constructions and the combinatorial theory of facevectors. The famous g-theorem describing integer sequences that can be the facevectors of simple (or simplicial) polytopes is one of the most striking applications oftoric geometry to combinatorics. The concepts of Gale duality and Gale diagramsare important tools for the study of moment-angle manifolds via intersections ofquadrics. In the additional sections we describe several combinatorial constructionsproviding families of simple polytopes, including nestohedra, graph associahedra,flagtopes and truncated cubes. The classical series of permutahedra and associahe-dra (Stasheff polytopes) are particular examples. The construction of nestohedratakes its origin in singularity and representation theory. We develop a differentialalgebraic formalism which links the generating series of nestohedra to classical par-tial differential equations. The potential of truncated cubes in toric topology is yetto be fully exploited, as they provide an immense source of explicitly constructedtoric spaces.

In Chapter 2 we describe systematically combinatorial structures that appearin the orbit spaces of toric objects. Besides convex polytopes, these include fans,simplicial and cubical complexes, and simplicial posets. All these structures areobjects of independent interest for combinatorialists, and we emphasised the aspectsof their combinatorial theory most relevant to subsequent topological applications.

The subject of Chapter 3 is the algebraic theory of face rings (also known asStanley–Reisner rings) of simplicial complexes, and their generalisations to simpli-cial posets. With the appearance of face rings at the beginning of the 1970s in thework of Reisner and Stanley, many combinatorial problems were translated into thelanguage of commutative algebra, which paved the way for their solution using theextensive machinery of algebraic and homological methods. Algebraic tools used forattacking combinatorial problems include regular sequences, Cohen–Macaulay andGorenstein rings, Tor-algebras, local cohomology, etc. A whole new thriving fieldappeared on the borders of combinatorics and algebra, which has since becomeknown as combinatorial commutative algebra.

Chapter 4 is the first ‘toric’ chapter of the book; it links the combinatorialand algebraic constructions of the previous chapters to the world of toric spaces.The concept of the moment-angle complex ZK is introduced as a functor from thecategory of simplicial complexes K to the category of topological spaces with torusactions and equivariant maps. When K is a triangulated manifold, the moment-angle complex ZK contains a free orbit Z∅ consisting of singular points. Removingthis orbit we obtain an open manifold ZK\Z∅, which satisfies the relative version of

xii INTRODUCTION

Poincare duality. Combinatorial invariants of simplicial complexes K therefore canbe described in terms of topological characteristics of the corresponding moment-angle complexes ZK. In particular, the face numbers of K, as well as the moresubtle bigraded Betti numbers of the face ring Z[K] can be expressed in terms of thecellular cohomology groups of ZK. The integral cohomology ring H∗(ZK) is shownto be isomorphic to the Tor-algebra TorZ[v1,...,vm](Z[K],Z). The proof builds upona construction of a ring model for cellular cochains of ZK and the correspond-ing cellular diagonal approximation, which is functorial with respect to maps ofmoment-angle complexes induced by simplicial maps of K. This functorial propertyof the cellular diagonal approximation for ZK is quite special, due to the lack ofsuch a construction for general cell complexes. Another result of Chapter 4 is a ho-motopy equivalence (an equivariant deformation retraction) from the complementU(K) of the arrangement of coordinate subspaces in Cm determined by K to themoment-angle complex ZK. Particular cases of this result are known in toric geom-etry and geometric invariant theory. It opens a new perspective on moment-anglecomplexes, linking them to the theory of configuration spaces and arrangements.

Toric varieties are the subject of Chapter 5. This is an extensive area witha vast literature. We outline the influence of toric geometry on the emergence oftoric topology and emphasise combinatorial, topological and symplectic aspectsof toric varieties. The construction of moment-angle manifolds via nondegenerateintersections of Hermitian quadrics in Cm, motivated by symplectic geometry, isalso discussed here. Some basic knowledge of algebraic geometry may be requiredin Chapter 5. Appropriate references are given in the introduction to the chapter.

The material of the first five chapters of the book should be accessible for agraduate student, or a reader with a very basic knowledge of algebra and topology.These five chapters may be also used for advanced courses on the relevant aspectsof topology, algebraic geometry and combinatorial algebra. The general algebraicand topological constructions required here are collected in Appendices A and Brespectively. The last four chapters are more research-oriented.

Geometry of moment-angle manifolds is studied in Chapter 6. The construc-tion of moment-angle manifolds as the level sets of toric moment maps is taken asthe starting point for the systematic study of intersections of Hermitian quadricsvia Gale duality. Following a remarkable discovery by Bosio and Meersseman ofcomplex-analytic structures on moment-angle manifolds corresponding to simplepolytopes, we proceed by showing that moment-angle manifolds corresponding toa more general class of complete simplicial fans can also be endowed with complex-analytic structures. The resulting family of non-Kahler complex manifolds includesthe classical series of Hopf and Calabi–Eckmann manifolds. We also describe im-portant invariants of these complex structures, such as the Hodge numbers andDolbeault cohomology rings, study holomorphic torus principal bundles over toricvarieties, and establish collapse results for the relevant spectral sequences. We con-clude by exploring the construction of A. E. Mironov providing a vast family ofLagrangian submanifolds with special minimality properties in complex space, com-plex projective space and other toric varieties. Like many other geometric construc-tions in this chapter, it builds upon the realisation of the moment-angle manifoldas an intersection of quadrics.

ACKNOWLEDGEMENTS xiii

In Chapter 7 we discuss several topological constructions of even-dimensionalmanifolds with an effective action of a torus of half the dimension of the mani-fold. They can be viewed as topological analogues and generalisations of compactnonsingular toric varieties (or toric manifolds). These include quasitoric manifoldsof Davis and Januszkiewicz, torus manifolds of Hattori and Masuda, and topologi-cal toric manifolds of Ishida, Fukukawa and Masuda. For all these classes of toricobjects, the equivariant topology of the action and the combinatorics of the orbitspaces interact in a harmonious way, leading to a host of results linking topologywith combinatorics. We also discuss the relationship with GKM-manifolds (namedafter Goresky, Kottwitz and MacPherson), another class of toric objects having itsorigin in symplectic topology.

Homotopy-theoretical aspects of toric topology are the subject of Chapter 8.This is now a very active area. Homotopy techniques brought to bear on the studyof polyhedral products and other toric spaces include model categories, homotopylimits and colimits, higher Whitehead and Samelson products. The required infor-mation about categorical methods in topology is collected in Appendix C.

In Chapter 9 we review applications of toric methods in a classical field ofalgebraic topology, complex cobordism. It is a generalised cohomology theory thatcombines both geometric intuition and elaborate algebraic techniques. The toricviewpoint brings an entirely new perspective on complex cobordism theory in bothits nonequivariant and equivariant versions.

The later chapters require more specific knowledge of algebraic topology, suchas characteristic classes and spectral sequences, for which we recommend respec-tively the classical book of Milnor and Stasheff [273] and the excellent guide byMcCleary [260]. Basic facts and constructions from bordism and cobordism theoryare given in Appendix D, while the related techniques of formal group laws andmultiplicative genera are reviewed in Appendix E.

Acknowledgements

We wish to express our deepest thanks to

· our teacher Sergei Petrovich Novikov for encouragement and support ofour research on toric topology;· Mikiya Masuda and Nigel Ray for long and much fruitful collaboration;· our coauthors in toric topology;· Peter Landweber for most helpful suggestions on improving the text;· all our colleagues who participated in conferences on toric topology forthe insight gained from discussions following talks and presentations.

This work was supported by the Russian Science Foundation (grant no. 14-11-00414). We also thank the Russian Foundation for Basic Research, the President ofthe Russian Federation Grants Council and Dmitri Zimin’s ‘Dynasty’ Foundationfor their support of our research related to this monograph.

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Index

Action (of group), 428almost free, 162, 429effective, 429free, 429proper, 195, 196, 216, 221semifree, 293, 429

transitive, 429Acyclic (space), 264, 419Adjoint functor, 439, 441Affine equivalence, 2A-genus, 394, 486Alexander duality, 67, 114Algebra, 395

bigraded, 396connected, 395exterior, 396finitely generated, 395graded, 395graded-commutative, 395

free, 396polynomial, 395multigraded, 396with straightening law (ASL), 116

Almost complex structure, 433associated with omniorientation, 247integrable, 433G-invariant, 381T -invariant, 247, 254, 434

Ample divisor, 187

Annihilator (of module), 408Arrangement (of subspaces), 95

coordinate, 157Aspherical (space), 415, 237Associahedron, 34

f -vector of, 52generalised, 25

Associated polyhedron (of intersection ofquadrics), 204

Atiyah–Hirzebruch formula, 374Augmentation, 418Augmentation genus, 485Axial function, 267, 304

n-independent, 3042-independent, 304

Baker–Akhiezer function, 488Bar construction, 448Barnette sphere, 71, 72, 184, 282Barycentre (of simplex), 64Barycentric subdivision

of polytope, 65

of simplicial complex, 64of simplicial poset, 83, 122

Base (of Schlegel diagram), 70Based map, 413Based space, 413Basepoint, 413Basis (of free module), 396Bernoulli number, 483Betti numbers (algebraic), 97, 120, 173Betti numbers (topological), 420

bigraded, 143Bidegree, 396Bistellar equivalence, 79Bistellar move, 78Blow-down, 310Blow-up

of T -graph, 310of T -manifold, 275

Boolean lattice, 81Bordism, 453

complex, 457oriented, 457unoriented, 453

Borel construction, 429Borel spectral sequence, 228Bott manifold, 288

generalised, 300Bott–Taubes polytope, 37Bott tower, 288

generalised, 300real, 301topologically trivial, 288

Boundary, 397Bounded flag, 284Bounded flag manifold, 284, 289, 361Bouquet (of spaces), 138, 413Bruckner sphere, 75Buchsbaum complex, 112

511

512 INDEX

Buchstaber invariant, 165

Building set, 29

graphical, 34

Calabi–Eckmann manifold, 230

Catalan number, 34

Categorical quotient, 188

CAT(0) inequality, 74

Cell complex, 414

Cellular approximation, 414

Cellular chain, 421

Cellular map, 414

Chain complex, 397, 442, 445

augmented, 418, 419

Chain homotopy, 398

Characteristic function, 244

directed, 247

Characteristic matrix, 247

refined, 248

Characteristic number, 464

Characteristic pair, 245

combinatorial, 247

equivalence of, 246, 247

Characteristic submanifold, 244, 261, 279

Charney–Davis Conjecture, 38, 74

Chern class, 463, 479

in complex cobordism, 464, 477

in equivariant cohomology, 432

in generalised cohomology theory, 484

of stably complex manifold, 464

Chern–Dold character, 365

Chern number, 464, 479

Chow ring, 185

CHP (covering homotopy property), 415

Classifying functor (algebraic), 448

Classifying space, 429

Clique, 65, 345

Cobar construction, 335, 448

Cobordism, 455, 456

complex, 458

equivariant

geometric, 357, 359

homotopic, 357, 358

unoriented, 457

Coboundary, 397

Cochain complex, 397, 442, 445

Cochain homotopy, 397

Cocycle, 397

Cofibrant (object, replacement,approximation), 440

Cofibration, 416

in model category, 440

Cofibre, 416

Coformality, 342, 443, 448

Cohen–Macaulay

algebra, 408

module, 408

simplicial complex, 106

simplicial poset, 121

Cohomological rigidity, 301

Cohomology

cellular, 422

of cochain complex, 397

simplicial, 418

reduced, 418

singular, 420

Cohomology product, 422

Cohomology product length, 156

Colimit, 172, 181, 314, 439

Combinatorial equivalence

of polyhedral complexes, 70

of polytopes, 2

Combinatorial neighbourhood, 63

Complex orientable map, 458

Complete intersection algebra, 111

Cone

over simplicial complex, 61

over space, 417

Cone (convex polyhedral), 56

dual, 56

rational, 56

regular, 56

simplicial, 56

strongly convex, 56

Connected sum

of manifolds, 344, 461

equivariant, 351

of simple polytopes, 6, 17

of simplicial complexes, 62

of stably complex manifolds, 462

Connection (on graph), 304

Contractible (space), 413

Contraction (of building set), 30

Coordinate subspace, 157

Coproduct, 439

Core (of simplicial complex), 63

Cross-polytope, 4

Cube, 2

standard, 2

topological, 83

Cubical complex

abstract, 83

polyhedral, 84

topological, 83

Cubical subdivision, 88

Cup product, 422

CW complex, 414

Cycle, 397

Cyclohedron, 37

Davis–Januszkiewicz space, 141, 316

Deformation retraction, 158

Degree (grading), 395

external, 399

internal, 399

total, 399

INDEX 513

Dehn–Sommerville relations

for polytopes, 15, 19

for simplicial complexes, 113

for simplicial posets, 127

for triangulated manifolds, 128

Depth (of module), 406

Derived functor, 441

Diagonal approximation, 145

Diagonal map, 422

Diagram (functor), 439

Diagram category, 439

Differential graded algebra (dg-algebra,dga), 398, 409, 442, 445

formal, 411

homologically connected, 409

minimal, 410

simply connected, 410

Differential graded coalgebra, 442, 445

Differential graded Lie algebra, 442, 447

Dimension

of module, 408

of polytope, 1

of simplicial complex, 59

of simplicial poset, 81

Discriminant (of elliptic curve), 475

Dolbeault cohomology, 225

Dolbeault complex, 225, 483

Double (simplicial), 165

Edge, 2

Eilenberg–MacLane space, 414

Eilenberg–Moore spectral sequence, 427

Elliptic cohomology, 488

Elliptic curve

Jacobi model, 475

Weierstrass model, 487

Elliptic formal group law, 475

universal, 475

Elliptic genus, 364, 392, 486, 489

universal, 486

Elliptic sine, 393, 475, 489

Equivariant bundle, 431

Equivariant characteristic class, 431

Equivariant cohomology, 431

of T -graph, 305

Equivariant map, 428

Euler class

in complex cobordism, 461, 477

in equivariant cobordism, 360

in equivariant cohomology, 261, 432

in generalised cohomology theory, 483

Euler formula, 15

Euler characteristic, 254, 377, 418, 419, 483

Eulerian poset, 127

Exact sequence, 397

of fibration, 415

of pair, 420

Excision, 420

Exponential (of formal group law), 474

Ext (functor), 404

Face

of cubical complex, 83

of cone, 56

of manifold with corners, 241, 263

of manifold with faces, 241

of polytope, 1

of simplicial complex, 59

of simplicial poset, 82

of T -graph, 305

Face category (of simplicial complex), 82,314, 439

Face coalgebra, 335

Face poset, 2

Face ring (Stanley–Reisner ring)

of manifold with corners, 265

of simple polytope, 92

of simplicial complex, 92

exterior, 142

of simplicial poset, 115

Facet, 2, 241, 305

Face truncation, 5

Fan, 56, 72

complete, 56

normal, 57

rational, 56

regular, 56

simplicial, 56

Fat wedge, 138

Fibrant (object, replacement,approximation), 440

Fibration, 415

in model category, 440

locally trivial, 413

Fibre bundle, 413

associated (with G-space), 429

Fixed point, 429

Fixed point data, 368

Flag complex (simplicial), 65, 340

Flag manifold, 382, 465

Flagtope (flag polytope), 38, 66

Folding map, 82

Formal group law, 473

Abel, 476

elliptic, 475

linearisable, 473

of geometric cobordisms, 477

universal, 475

Formality

in model category, 443

integral, 316

of dg-algebra, 411

of space 168, 315, 320, 425

F -polynomial (of polytope), 14

Frolicher spectral sequence, 228

Full subcomplex (of simplicial complex), 63

514 INDEX

Fundamental group, 413

Fundamental homology class, 156

f -vector (face vector)

of cubical complex, 84

of polytope, 14

of simplicial complex, 60

of simplicial poset, 117

Gal Conjecture, 39, 74

Gale diagram, 11, 208, 236

combinatorial, 13, 207

Gale duality, 10, 204, 210, 212, 227

Gale transform, 10

Ganea’s Theorem, 143

g-conjecture, 73, 113

Generalised (co)homology theory, 454,

complex oriented, 483

multiplicative, 457

Generalised Lower Bound Conjecture(GLBC), 22, 188

Generating series

of face polynomials, 51

of polytopes, 49

Genus, 364, 479

equivariant, 365

fibre multiplicative, 367

oriented, 486

rigid, 366

universal, 482

Geometric cobordism, 461

Geometric quotient, 189

Geometric realisation

of cubical complex, 84

of simplicial complex, 59

of simplicial set, 442

Ghost vertex, 59

GKM-graph, 303

GKM-manifold, 303

Golod (ring, simplicial complex), 171, 345

Gorenstein, Gorenstein*

simplicial complex, 112, 154

simplicial poset, 125

Goresky–MacPherson formula, 161

Graded Lie algebra, 447

Graph

chordal, 345

of polytope, 9

simple, 9, 34

Graph-associahedron, 34

Graph product, 341

Grassmannian, 188, 382

g-theorem, 23, 187

g-vector

of polytope, 14

of simplicial complex, 60

Gysin homomorphism

in equivariant cohomology, 261, 432

in cobordism, 361, 460

Gysin–Thom isomorphism, 369, 463

Half-smash product (left, right), 322

Hamiltonian action, 195, 293

Hamiltonian-minimal submanifold, 231

Hamiltonian vector field, 231

Hard Lefschetz Theorem, 24, 187

Hauptvermutung, 76

Heisenberg group, 426

HEP (homotopy extension property), 416

Hermitian quadric, 197, 202

Hilton–Milnor Theorem, 138

Hirzebruch genus, 364, 480

equivariant, 365

fibre multiplicative, 366

oriented, 486

rigid, 364, 366, 385

universal, 482

Hirzebruch surface, 182

H-minimal submanifold, 231

Hodge algebra, 116

Hodge number, 225

Homological dimension (of module), 399

Homology

cellular, 421

of chain complex, 397

simplicial, 418

reduced, 418

singular, 419

of pair, 420

reduced, 419

Homology polytope, 264

Homology sphere, 76

Homotopy, 413

Homotopy category (of model category),441

Homotopy cofibre, 417

Homotopy colimit, 314, 450

Homotopy equivalence, 413

Homotopy fibre, 416

Homotopy group, 413

Homotopy Lie algebra, 342, 423, 443

Homotopy limit, 314, 450

Homotopy quotient, 429

Homotopy type, 413

Hopf Conjecture, 74

Hopf equation, 50

Hopf line bundle, 430

Hopf manifold, 222

H-polynomial (of polytope), 14

hsop (homogeneous system of parameters),408

in face rings, 105, 117

Hurewicz homomorphism, 424, 467

in complex cobordism, 466

Hurwitz series, 474, 489

h-vector

of polytope, 14

INDEX 515

of simplicial complex, 60

of simplicial poset, 117

Hyperplane cut, 5

Intersection homology, 24, 187

Intersection of quadrics, 197, 202

nondegenerate (transverse), 202

Intersection poset (of arrangement), 161Isotropy representation, 381, 430

Join (least common upper bound), 82, 114

Join (operation)

of simplicial complexes, 61

of simplicial posets, 172

of spaces, 322

Klein bottle, 234

Koszul algebra, 403Koszul complex, 403

Koszul resolution, 400

Krichever genus, 389, 489

universal, 389, 492

K-theory, 485

Lagrangian immersion, 231

Lagrangian submanifold, 231

Landweber Exact Functor Theorem, 484

Latching functor, 444

Lattice, 180, 210Lefschetz pair, 134

Left lifting property, 416

Leibniz identity, 398

L-genus (signature), 483

Limit (of diagram), 439

Link (in simplicial complex), 62, 121

Link (of intersection of quadrics), 207

Linkage, 130

Localisation formula, 368

Locally standard (torus action), 240, 257

Logarithm (of formal group law), 474

Loop functor (algebraic), 448

Loop space, 331, 415Lower Bound Theorem (LBT), 22

lsop (linear system of parameters), 408

in face rings, 105, 117

integral, 106

LVM-manifold, 213, 223

Manifold with corners, 133, 241

face-acyclic, 264

nice, 241

Manifold with faces, 241

Mapping cone, 417

Mapping cylinder, 417Massey product, 168, 411

indecomposable, 171

indeterminacy of, 412

trivial (vanishing), 412

Matching functor, 444

Mayer–Vietoris sequence, 420

Maximal action (of torus), 223Meet (greatest common lower bound), 82,

114

Milnor hypersurface, 348, 469Minimal basis (of graded module), 400Minimal model

of dg-algebra, 410, 443of space, 425

Minimal submanifold, 231Minkowski sum, 26

Missing face, 65, 92, 332, 340Model (of dg-algebra), 410

Model category, 440Module, 395

finitely generated, 395free, 396graded, 395

projective, 396Moment-angle complex, 131, 172

real, 134, 149Moment-angle manifold, 134, 197

polytopal, 134, 206non-formal, 169

Moment map, 195, 209proper, 196, 210

Monoid, 136

Monomial ideal, 92Moore loops, 331

Morava K-theory, 473Multi-fan, 239, 380

Multidegree, 396Multigrading, 100, 148, 173

M -vector, 23

Nerve complex, 60, 191Nested set, 31Nestohedron, 31

Nilpotent space, 424Non-PL sphere, 73, 76

Normal complex structure, 433

Octahedron, 4Omniorientation, 247, 261, 279, 309

Opposite category, 439Orbifold, 183

Orbit (of group action), 428Orbit space, 429

Order complex (of poset), 65Overcategory, 440

Partition, 465Path space, 415

Pair (of spaces), 413Perfect elimination order, 345

Permutahedron, 28Picard group, 195

PL map, 61homeomorphism, 61

516 INDEX

PL manifold, 76

PL sphere, 69

Poincare algebra, 112

Poincare–Atiyah duality, 457, 460

Poincare duality, 421

Poincare duality space, 154

Poincare series 341, 399

of face ring, 94

Poincare sphere, 76

Pointed map, 413

Pointed space, 413

Polar set, 3

Polyhedral complex, 70

Polyhedral product, 135, 172, 313

Polyhedral smash product, 326

Polyhedron (convex), 1

Delzant, 210

Polyhedron (simplicial complex), 59

Polytopal sphere, 72

Polytope

combinatorial, 2

convex, 1

cyclic, 7

Delzant, 57, 184, 199

dual, 4

generic, 3

lattice, 183

neighbourly, 7, 17

nonrational, 185

polar, 3

regular, 4

self-dual, 4

simple, 3, 46

simplicial, 3, 46

stacked, 22

triangle-free, 43

Polytope algebra, 24

Pontryagin algebra, 331

Pontryagin class, 486

Pontryagin number, 464

Pontryagin product, 423

Pontryagin–Thom map, 432, 454

Poset (partially ordered set), 2

Poset category, 439

Positive orthant, 3, 9, 131

Presentation (of a polytope byinequalities), 2

generic, 3

irredundant, 2

rational, 210

Primitive (lattice vector), 56

Principal bundle, 429

Principal minor (of matrix), 291

Product

of building sets, 47

of polytopes, 5

Product (categorical), 439

Products (in cobordism), 459

Projective dimension (of module), 399

Projectivisation (of vector bundle), 463Pseudomanifold, 152, 307

orientable, 153Pullback, 439Pushout, 439

Quadratic algebra, 92, 340

Quasi-isomorphism, 398, 410, 442Quasitoric manifold, 244, 318, 376

almost complex structure on, 254canonical smooth structure on, 249

complex structure on, 254equivalence of, 246, 247

Quillen pair (of functors), 441Quotient space, 429

Rank (of free module), 396Rank function, 81

Rational equivalence, 424Rational homotopy type, 424

Ray’s basis, 363Redundant inequality, 2

Reedy category, 444, 450Regular sequence, 405Regular subdivision, 121

Regular value (of moment map), 196, 210Reisner Theorem, 107

Resolution (of module)free, 399

minimal, 400projective, 399

Restriction (of a building set), 26Restriction map

algebraic, 94, 105, 116, 266

in equivariant cohomology, 259Right-angled Artin group, 341

Right-angled Coxeter group, 341Right lifting property, 416

Rigidity equation, 386Ring of polytopes, 44

Root (of representation), 381complementary, 381of almost complex structure, 381

Samelson product, 332, 423, 443

higher, 333Schlegel diagram, 70

Segre embedding, 469Seifert fibration, 225

Sign (of fixed point), 251, 377, 434Signature, 377, 483Simplex, 2, 82

abstract, 59regular, 2

standard, 2Simplicial cell complex, 82

Simplicial chain, 417Simplicial cochain, 418

INDEX 517

Simplicial complex

abstract, 59

Alexander dual, 66

geometric, 59

Golod, 345

neighbourly, 141

pure, 59

shifted, 171, 325

Simplicial manifold, 76

Simplicial map, 60

isomorphism, 60

nondegenerate, 60

Simplicial object (in category), 439

Simplicial poset, 81

dual (of manifold with corners), 265

Simplicial set, 439

Simplicial sphere, 69

Simplicial subdivistion, 61

Simplicial wedge, 165

Singular chain, 419

of pair, 420

Singular cochain, 420

Skeleton (of cell complex), 414

Slice Theorem, 430

Small category, 439

Small cover, 234, 283, 301

Smash product, 322, 413

Special unitary (SU-) manifold, 389

Stabiliser (of group action), 428

Stably complex structure, 250, 433

T -invariant, 434

Stanley–Reisner ideal, 92

Star, 62, 121, 218

Starshaped sphere, 72, 221

Stasheff polytope, 25

Stationary subgroup, 428

Steinitz Problem, 75

Steinitz Theorem, 71, 72

Stellahedron, 37

Stellar subdivision

of simplicial complex, 78

of simplicial poset, 122

Stiefel–Whitney class, 464

in equivariant cohomology, 432

Stiefel–Whitney number, 464

Straightening relation, 116

Subcomplex of simplicial complex, 59

Sullivan algebra (of piecewise polynomialdifferential forms), 424

Supporting hyperplane, 1

Suspension

of module, 396

of simplicial complex, 61

of space, 417

Suspension isomorphism, 421

Symplectic manifold, 195

Symplectic reduction, 196, 209, 237

Symplectic quotient, 196, 209

Syzygy, 399

Tangent bundle along the fibres, 250, 431

Tangential representation, 430

Tautological line bundle, 430

over projectivisation, 463

Tensor product (of modules), 396

T -graph (torus graph), 304

Thom class, 267, 305, 432

Thom isomorphism, 467

Thom space, 454

T -manifold, 239

Todd genus (td), 378, 483

Topological fan, 280

complete, 280

Topological monoid, 442

Topological toric manifold, 278

Tor (functor), 401

Tor-algebra, 97

Toral rank, 162

Toric manifold, 185, 237, 242

Hamiltonian, 199, 212

over cube, 288

projective, 185

Toric space, xi, 130

Toric variety, 24, 179

affine, 180

projective, 183

real, 234, 283

Torus

algebraic, 179

standard, 130

Torus manifold, 257

Triangulated manifold, 76

Triangulated sphere, 21, 69

flag, 74

Triangulation, 55, 61

neighbourly, 75, 141

Tropical geometry, 283

Truncated cube, 40

Truncated simplex, 38

Truncation, 5

Type (of S1-action), 389

Undercategory, 440

Underlying simplicial complex (of fan), 61

Unipotent (upper triangular matrix), 291

Universal G-space (universal bundle), 429

Universal toric genus, 357, 360

Upper Bound Theorem (UBT), 20

Vector bundle, 429

Vertex

of polytope, 2

of simplicial complex, 59

of simplicial poset, 81

Vertex truncation (vt), 5

Volume polynomial, 24

518 INDEX

Weak equivalenceof dg-algebras, 410in model category, 440

Wedge (of spaces), 138, 413of spheres, 171, 321, 345

Weierstrass ℘-function, 387, 487Weierstrass σ-function, 487, 488

Weight (of torus representation), 250, 377,434

Weight graph, 302Weight lattice (of torus), 210, 434Weyl group, 380Whitehead product, 422

higher, 332Witten genus, 487

Yoneda algebra, 448

Zigzag (of maps), 410Zonotope, 29

γ-polynomial, 18γ-vector

of polytope, 18of triangulated sphere, 74

χa,b-genus, 373, 483χy-genus, 373, 483

2-truncated cube, 4024-cell, 45-lemma, 398

SURV/204

This book is about toric topology, a new area of mathematics that emerged at the end of the 1990s on the border of equivariant topology, algebraic and symplectic geometry, combina-torics, and commutative algebra. It has quickly grown into a very active area with many links to other areas of mathematics, and continues to attract experts from different fields.

The key players in toric topology are moment-angle manifolds, a class of manifolds with torus actions defined in combinatorial terms. Construction of moment-angle manifolds relates to combinatorial geometry and alge-braic geometry of toric varieties via the notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to important connec-tions with classical and modern areas of symplectic, Lagrangian, and non-Kaehler complex geometry. A related categorical construction of moment-angle complexes and polyhedral products provides for a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate subject of homotopy theory. A new perspective on torus actions has also contributed to the development of classical areas of algebraic topology, such as complex cobordism.

This book includes many open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter this beautiful new area.

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