Mathematical Surveys and Monographs Volume 204

Toric

Victor M. Buchstaber Taras E. Panov

American Mathematical Society Toric Topology

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

Mathematical Surveys and Monographs Volume 204

Toric Topology

Victor M. Buchstaber Taras E. Panov

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Constantin Teleman Michael A. Singer, Chair MichaelI.Weinstein Benjamin Sudakov

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

For additional information and updates on this book, visit www.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. . 4. , Algebraic. I. Panov, Taras E., 1975– II. Title

QA613.2.B82 2015 516.35–dc23 2015006771

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Introduction ix Chapter guide x Acknowledgements xiii

Chapter 1. Geometry and of Polytopes 1 1.1. Convex polytopes 1 1.2. Gale and Gale diagrams 9 1.3. Face vectors and Dehn–Sommerville relations 14 1.4. Characterising the face vectors of polytopes 19 Polytopes: Additional Topics 25 1.5. Nestohedra and graph-associahedra 25 1.6. Flagtopes and truncated cubes 38 1.7. Differential algebra of combinatorial polytopes 44 1.8. Families of polytopes and differential equations 48

Chapter 2. Combinatorial Structures 55 2.1. Polyhedral fans 56 2.2. Simplicial complexes 59 2.3. Barycentric subdivision and flag complexes 64 2.4. 66 2.5. Classes of triangulated spheres 69 2.6. Triangulated 75 2.7. Stellar subdivisions and bistellar moves 78 2.8. Simplicial posets and simplicial cell complexes 81 2.9. Cubical complexes 83

Chapter 3. Combinatorial Algebra of Face Rings 91 3.1. Face rings of simplicial complexes 92 3.2. Tor-algebras and Betti numbers 97 3.3. Cohen–Macaulay complexes 104 3.4. Gorenstein complexes and Dehn–Sommerville relations 111 3.5. Face rings of simplicial posets 114 Face Rings: Additional Topics 121 3.6. Cohen–Macaulay simplicial posets 121 3.7. Gorenstein simplicial posets 125 3.8. Generalised Dehn–Sommerville relations 127

Chapter 4. Moment-Angle Complexes 129 4.1. Basic definitions 131 4.2. Polyhedral products 135

v vi CONTENTS

4.3. Homotopical properties 139 4.4. Cell decomposition 143 4.5. ring 144 4.6. Bigraded Betti numbers 150 4.7. Coordinate subspace arrangements 157 Moment-Angle Complexes: Additional Topics 162 4.8. Free and almost free torus actions on moment-angle complexes 162 4.9. Massey products in the cohomology of moment-angle complexes 168 4.10. Moment-angle complexes from simplicial posets 171

Chapter 5. Toric Varieties and Manifolds 179 5.1. Classical construction from rational fans 179 5.2. Projective toric varieties and polytopes 183 5.3. Cohomology of toric manifolds 185 5.4. Algebraic quotient construction 188 5.5. Hamiltonian actions and symplectic reduction 195

Chapter 6. Geometric Structures on Moment-Angle Manifolds 201 6.1. Intersections of quadrics 201 6.2. Moment-angle manifolds from polytopes 205 6.3. Symplectic reduction and moment maps revisited 209 6.4. Complex structures on intersections of quadrics 212 6.5. Moment-angle manifolds from simplicial fans 215 6.6. Complex structures on moment-angle manifolds 219 6.7. Holomorphic principal bundles and 224 6.8. Hamiltonian-minimal Lagrangian submanifolds 231

Chapter 7. Half-Dimensional Torus Actions 239 7.1. Locally standard actions and manifolds with corners 240 7.2. Toric manifolds and their quotients 242 7.3. Quasitoric manifolds 243 7.4. Locally standard T -manifolds and torus manifolds 257 7.5. Topological toric manifolds 278 7.6. Relationship between different classes of T -manifolds 282 7.7. Bounded flag manifolds 284 7.8. Bott towers 287 7.9. Weight graphs 302

Chapter 8. Theory of Polyhedral Products 313 8.1. of polyhedral products 314 8.2. Wedges of spheres and connected sums of sphere products 321 8.3. Stable decompositions of polyhedral products 326 8.4. Loop spaces, Whitehead and Samelson products 331 8.5. The case of flag complexes 340

Chapter 9. Torus Actions and Complex 347 9.1. Toric and quasitoric representatives in complex bordism classes 347 9.2. The universal toric genus 357 9.3. Equivariant genera, rigidity and fibre multiplicativity 364 9.4. Isolated fixed points: localisation formulae 367 CONTENTS vii

9.5. Quasitoric manifolds and genera 376 9.6. Genera for homogeneous spaces of compact Lie groups 380 9.7. Rigid genera and functional equations 385 Appendix A. Commutative and 395 A.1. Algebras and modules 395 A.2. Homological theory of graded rings and modules 398 A.3. Regular sequences and Cohen–Macaulay algebras 405 A.4. Formality and Massey products 409 Appendix B. Algebraic Topology 413 B.1. Homotopy and 413 B.2. Elements of rational homotopy theory 424 B.3. Eilenberg–Moore spectral sequences 426 B.4. Group actions and equivariant topology 428 B.5. Stably complex structures 433 B.6. Weights and signs of torus actions 434

Appendix C. Categorical Constructions 439 C.1. Diagrams and model categories 439 C.2. Algebraic model categories 444 C.3. Homotopy limits and colimits 450 Appendix D. Bordism and Cobordism 453 D.1. Bordism of manifolds 453 D.2. Thom spaces and cobordism functors 454 D.3. Oriented and complex bordism 457 D.4. Characteristic classes and numbers 463 D.5. Structure results 467 D.6. Ring generators 468 Appendix E. Formal Group Laws and Hirzebruch Genera 473 E.1. Elements of the theory of formal group laws 473 E.2. Formal group law of geometric 477 E.3. Hirzebruch genera (complex case) 479 E.4. Hirzebruch genera (oriented case) 486 E.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 torus actions arise in different areas of mathematics and , which re- sults in permanent interest in the theory, new applications and penetration of new ideas into topology. Since the 1970s, algebraic and symplectic viewpoints on torus actions have enriched the subject with new combinatorial ideas and methods, largely based on the convex-geometric concept of polytopes. The study of algebraic torus actions on algebraic varieties has quickly devel- oped into a highly successful branch of , known as toric geometry. It gives a bijection between, on the one hand, toric varieties, which are complex algebraic 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 allows one to completely translate various algebraic-geometric notions into combinatorics. Projective toric varieties correspond to fans which arise from convex polytopes. A valuable aspect of this theory is that it provides many explicit examples of alge- braic varieties, leading to applications in deep subjects such as singularity theory and mirror symmetry. In symplectic geometry, since the early 1980s there has been much activity in the field of Hamiltonian group actions on symplectic manifolds. Such an action defines the moment map from the to a Euclidean space (more precisely, the dual Lie algebra of the torus) whose image is a convex polytope. If the torus has half the dimension of the manifold, the image of the moment map determines the manifold up to equivariant symplectomorphism. The class of polytopes which arise as the images of moment maps can be described explicitly, together with an effective procedure for recovering a symplectic manifold from such a polytope. In symplectic geometry, as in algebraic geometry, one translates various geometric constructions into 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 a moment map. The image of the moment map is a convex polytope that is dual to the fan. In both the smooth algebraic-geometric and the symplectic situations, the compact 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 action is naturally a manifold with corners, stratified according to the dimension of the stabilisers, and each stratum can be equipped with data that encodes the isotropy torus action along that stratum. Not only does this structure of the quotient provide a powerful means of investigating the action, but some of its subtler combinatorial properties 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 the 1990s, the ideas and methodology of toric varieties and Hamiltonian torus actions have started penetrating back into algebraic topology. By 2000, several constructions of topological analogues of toric varieties and symplectic toric manifolds had appeared in the literature, together with different seemingly unrelated realisations of what later has become known as moment-angle manifolds. We tried to systematise both known and emerging links between torus actions and combinatorics in our 2000 paper [67] in Russian Mathematical Sur- veys, where the terms ‘moment-angle manifold’ and ‘moment-angle complex’ first appeared. Two years later it grew into a book Torus Actions and Their Applications in Topology and Combinatorics [68] published by the AMS in 2002 (the extended Russian edition [69] appeared in 2004). The title ‘Toric Topology’ coined by our colleague Nigel Ray became official after the 2006 Osaka conference under the same name. Its proceedings volume [177] contained many important contributions to the subject, 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 many young researchers entering the subject and conferences held around the world every year, toric topology has definitely grown into a mature area. Its various aspects are presented in this monograph, with an intention to consolidate the foundations and stimulate further applications.

Chapter guide

1 2 3 - - Polytopes Combinatorial Face structures rings

? ?

5 4 8 - Toric Moment-angle Homotopy varieties complexes theory Z  Z  Z ? = ZZ~ ?

6 7 9 - Moment-angle Half-dim Cobordism manifolds torus actions

Each chapter and most sections have their own introductions with more specific information about the contents. ‘Additional topics’ of Chapters 1, 3 and 4 contain more 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 to them only for notation and terminology. At the heart of toric topology lies a class of torus actions whose orbit spaces are highly structured in combinatorial terms, that is, have lots of orbit types tied together in a nice combinatorial way. We use the generic terms toric space and toric object to refer to a with a nice torus action, or to a space produced from a torus action via different standard topological or categorical constructions. Examples of toric spaces include toric varieties, toric and quasitoric manifolds and their generalisations, moment-angle manifolds, moment-angle complexes and their Borel 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 face vectors. The famous g-theorem describing integer sequences that can be the face vectors of simple (or simplicial) polytopes is one of the most striking applications of toric geometry to combinatorics. The concepts of Gale duality and Gale diagrams are important tools for the study of moment-angle manifolds via intersections of quadrics. In the additional sections we describe several combinatorial constructions providing 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 nestohedra takes its origin in singularity and representation theory. We develop a differential algebraic formalism which links the generating series of nestohedra to classical par- tial differential equations. The potential of truncated cubes in toric topology is yet to be fully exploited, as they provide an immense source of explicitly constructed toric spaces. In Chapter 2 we describe systematically combinatorial structures that appear in the orbit spaces of toric objects. Besides convex polytopes, these include fans, simplicial and cubical complexes, and simplicial posets. All these structures are objects of independent interest for combinatorialists, and we emphasised the aspects of their combinatorial theory most relevant to subsequent topological applications. The subject of Chapter 3 is the algebraic theory of face rings (also known as Stanley–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 the work of Reisner and Stanley, many combinatorial problems were translated into the language of commutative algebra, which paved the way for their solution using the extensive machinery of algebraic and homological methods. Algebraic tools used for attacking combinatorial problems include regular sequences, Cohen–Macaulay and Gorenstein rings, Tor-algebras, local cohomology, etc. A whole new thriving field appeared on the borders of combinatorics and algebra, which has since become known as combinatorial commutative algebra. Chapter 4 is the first ‘toric’ chapter of the book; it links the combinatorial and 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 the category of simplicial complexes K to the category of topological spaces with torus actions and equivariant maps. When K is a triangulated manifold, the moment- angle complex ZK contains a free orbit Z∅ consisting of singular points. Removing this orbit we obtain an open manifold ZK \Z∅, which satisfies the relative version of xii INTRODUCTION

Poincar´e duality. Combinatorial invariants of simplicial complexes K therefore can be described in terms of topological characteristics of the corresponding moment- angle complexes ZK. In particular, the face numbers of K,aswellasthemore subtle bigraded Betti numbers of the face ring Z[K] can be expressed in terms of the ∗ cellular cohomology groups of ZK. The integral cohomology ring H (ZK)isshown Z K Z to be isomorphic to the Tor-algebra TorZ[v1,...,vm]( [ ], ). The proof builds upon a construction of a ring model for cellular cochains of ZK and the correspond- ing cellular diagonal approximation, which is functorial with respect to maps of moment-angle complexes induced by simplicial maps of K. This functorial property of the cellular diagonal approximation for ZK is quite special, due to the lack of such a construction for general cell complexes. Another result of Chapter 4 is a ho- motopy equivalence (an equivariant deformation retraction) from the complement U(K) of the arrangement of coordinate subspaces in Cm determined by K to the moment-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-angle complexes, linking them to the theory of configuration spaces and arrangements. Toric varieties are the subject of Chapter 5. This is an extensive area with a vast literature. We outline the influence of toric geometry on the emergence of toric topology and emphasise combinatorial, topological and symplectic aspects of toric varieties. The construction of moment-angle manifolds via nondegenerate intersections of Hermitian quadrics in Cm, motivated by symplectic geometry, is also discussed here. Some basic knowledge of algebraic geometry may be required in 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 a graduate 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 aspects of topology, algebraic geometry and combinatorial algebra. The general algebraic and topological constructions required here are collected in Appendices A and B respectively. 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 as the starting point for the systematic study of intersections of Hermitian quadrics via Gale duality. Following a remarkable discovery by Bosio and Meersseman of complex-analytic structures on moment-angle manifolds corresponding to simple polytopes, we proceed by showing that moment-angle manifolds corresponding to a more general class of complete simplicial fans can also be endowed with complex- analytic structures. The resulting family of non-K¨ahler complex manifolds includes the classical series of Hopf and Calabi–Eckmann manifolds. We also describe im- portant invariants of these complex structures, such as the Hodge numbers and Dolbeault cohomology rings, study holomorphic torus principal bundles over toric varieties, and establish collapse results for the relevant spectral sequences. We con- clude by exploring the construction of A. E. Mironov providing a vast family of Lagrangian 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 manifold as an intersection of quadrics. ACKNOWLEDGEMENTS xiii

In Chapter 7 we discuss several topological constructions of even-dimensional manifolds 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 compact nonsingular toric varieties (or toric manifolds). These include quasitoric manifolds of Davis and Januszkiewicz, torus manifolds of Hattori and Masuda, and topologi- cal toric manifolds of Ishida, Fukukawa and Masuda. For all these classes of toric objects, the equivariant topology of the action and the combinatorics of the orbit spaces interact in a harmonious way, leading to a host of results linking topology with combinatorics. We also discuss the relationship with GKM-manifolds (named after Goresky, Kottwitz and MacPherson), another class of toric objects having its origin 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 study of polyhedral products and other toric spaces include model categories, homotopy limits 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 of algebraic topology, . It is a generalised cohomology theory that combines both geometric intuition and elaborate algebraic techniques. The toric viewpoint brings an entirely new perspective on complex cobordism theory in both its nonequivariant and equivariant versions. The later chapters require more specific knowledge of algebraic topology, such as characteristic classes and spectral sequences, for which we recommend respec- tively the classical book of Milnor and Stasheff [273] and the excellent guide by McCleary [260]. Basic facts and constructions from bordism and cobordism theory are given in Appendix D, while the related techniques of formal group laws and multiplicative genera are reviewed in Appendix E.

Acknowledgements We wish to express our deepest thanks to · our teacher Sergei Petrovich Novikov for encouragement and support of our 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 for the 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 of the Russian Federation Grants Council and Dmitri Zimin’s ‘Dynasty’ Foundation for their support of our research related to this monograph.

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Index

Action (of group), 428 Baker–Akhiezer function, 488 almost free, 162, 429 Bar construction, 448 effective, 429 Barnette sphere, 71, 72, 184, 282 free, 429 Barycentre (of simplex), 64 proper, 195, 196, 216, 221 Barycentric subdivision semifree, 293, 429 of polytope, 65 transitive, 429 of , 64 Acyclic (space), 264, 419 of simplicial poset, 83, 122 Adjoint functor, 439, 441 Base (of Schlegel diagram), 70 Affine equivalence, 2 Based map, 413 A-genus, 394, 486 Based space, 413 Alexander duality, 67, 114 Basepoint, 413 Algebra, 395 Basis (of free module), 396 bigraded, 396 Bernoulli number, 483 connected, 395 Betti numbers (algebraic), 97, 120, 173 exterior, 396 Betti numbers (topological), 420 finitely generated, 395 bigraded, 143 graded, 395 Bidegree, 396 graded-commutative, 395 Bistellar equivalence, 79 free, 396 Bistellar move, 78 polynomial, 395 Blow-down, 310 multigraded, 396 Blow-up with straightening law (ASL), 116 of T -graph, 310 Almost complex structure, 433 of T -manifold, 275 associated with omniorientation, 247 Boolean lattice, 81 integrable, 433 Bordism, 453 G-invariant, 381 complex, 457 T -invariant, 247, 254, 434 oriented, 457 Ample divisor, 187 unoriented, 453 Annihilator (of module), 408 Borel construction, 429 Arrangement (of subspaces), 95 Borel spectral sequence, 228 coordinate, 157 Bott manifold, 288 Aspherical (space), 415, 237 generalised, 300 Associahedron, 34 Bott–Taubes polytope, 37 f-vector of, 52 Bott tower, 288 generalised, 25 generalised, 300 Associated polyhedron (of intersection of real, 301 quadrics), 204 topologically trivial, 288 Atiyah–Hirzebruch formula, 374 Boundary, 397 Augmentation, 418 Bounded flag, 284 Augmentation genus, 485 Bounded flag manifold, 284, 289, 361 Axial function, 267, 304 Bouquet (of spaces), 138, 413 n-independent, 304 Br¨uckner sphere, 75 2-independent, 304 Buchsbaum complex, 112

511 512 INDEX

Buchstaber invariant, 165 simplicial poset, 121 Building set, 29 Cohomological rigidity, 301 graphical, 34 Cohomology cellular, 422 Calabi–Eckmann manifold, 230 of cochain complex, 397 Catalan number, 34 simplicial, 418 Categorical quotient, 188 reduced, 418 CAT(0) inequality, 74 singular, 420 Cell complex, 414 Cohomology product, 422 Cellular approximation, 414 Cohomology product length, 156 Cellular chain, 421 Colimit, 172, 181, 314, 439 Cellular map, 414 Combinatorial equivalence Chain complex, 397, 442, 445 of polyhedral complexes, 70 augmented, 418, 419 of polytopes, 2 Chain homotopy, 398 Combinatorial neighbourhood, 63 Characteristic function, 244 Complex orientable map, 458 directed, 247 Complete intersection algebra, 111 Characteristic matrix, 247 Cone refined, 248 over simplicial complex, 61 Characteristic number, 464 over space, 417 Characteristic pair, 245 Cone (convex polyhedral), 56 combinatorial, 247 dual, 56 equivalence of, 246, 247 rational, 56 Characteristic submanifold, 244, 261, 279 regular, 56 Charney–Davis Conjecture, 38, 74 simplicial, 56 , 463, 479 strongly convex, 56 in complex cobordism, 464, 477 Connected sum in equivariant cohomology, 432 of manifolds, 344, 461 in generalised cohomology theory, 484 equivariant, 351 of stably complex manifold, 464 of simple polytopes, 6, 17 Chern–Dold character, 365 of simplicial complexes, 62 Chern number, 464, 479 of stably complex manifolds, 462 Chow ring, 185 Connection (on graph), 304 CHP (covering homotopy property), 415 Contractible (space), 413 Classifying functor (algebraic), 448 Contraction (of building set), 30 Classifying space, 429 Coordinate subspace, 157 Clique, 65, 345 Coproduct, 439 Cobar construction, 335, 448 Core (of simplicial complex), 63 Cobordism, 455, 456 Cross-polytope, 4 complex, 458 Cube, 2 equivariant standard, 2 geometric, 357, 359 topological, 83 homotopic, 357, 358 unoriented, 457 abstract, 83 Coboundary, 397 polyhedral, 84 Cochain complex, 397, 442, 445 topological, 83 Cochain homotopy, 397 Cubical subdivision, 88 Cocycle, 397 , 422 Cofibrant (object, replacement, CW complex, 414 approximation), 440 Cycle, 397 Cofibration, 416 Cyclohedron, 37 in model category, 440 Cofibre, 416 Davis–Januszkiewicz space, 141, 316 Coformality, 342, 443, 448 Deformation retraction, 158 Cohen–Macaulay Degree (grading), 395 algebra, 408 external, 399 module, 408 internal, 399 simplicial complex, 106 total, 399 INDEX 513

Dehn–Sommerville relations Exponential (of formal group law), 474 for polytopes, 15, 19 Ext (functor), 404 for simplicial complexes, 113 for simplicial posets, 127 Face for triangulated manifolds, 128 of cubical complex, 83 Depth (of module), 406 of cone, 56 Derived functor, 441 of manifold with corners, 241, 263 Diagonal approximation, 145 of manifold with faces, 241 Diagonal map, 422 of polytope, 1 Diagram (functor), 439 of simplicial complex, 59 Diagram category, 439 of simplicial poset, 82 Differential graded algebra (dg-algebra, of T -graph, 305 dga), 398, 409, 442, 445 Face category (of simplicial complex), 82, formal, 411 314, 439 homologically connected, 409 Face coalgebra, 335 minimal, 410 Face poset, 2 simply connected, 410 Face ring (Stanley–Reisner ring) Differential graded coalgebra, 442, 445 of manifold with corners, 265 Differential graded Lie algebra, 442, 447 of simple polytope, 92 Dimension of simplicial complex, 92 of module, 408 exterior, 142 of polytope, 1 of simplicial poset, 115 of simplicial complex, 59 Facet, 2, 241, 305 of simplicial poset, 81 Face truncation, 5 Discriminant (of elliptic curve), 475 Fan, 56, 72 Dolbeault cohomology, 225 complete, 56 Dolbeault complex, 225, 483 normal, 57 Double (simplicial), 165 rational, 56 regular, 56 Edge, 2 simplicial, 56 Eilenberg–Mac Lane space, 414 Fat wedge, 138 Eilenberg–Moore spectral sequence, 427 Fibrant (object, replacement, Elliptic cohomology, 488 approximation), 440 Elliptic curve Fibration, 415 Jacobi model, 475 in model category, 440 Weierstrass model, 487 locally trivial, 413 Elliptic formal group law, 475 Fibre bundle, 413 universal, 475 associated (with G-space), 429 Elliptic genus, 364, 392, 486, 489 Fixed point, 429 universal, 486 Fixed point data, 368 Elliptic sine, 393, 475, 489 Flag complex (simplicial), 65, 340 Equivariant bundle, 431 Flag manifold, 382, 465 Equivariant , 431 Flagtope (flag polytope), 38, 66 Equivariant cohomology, 431 Folding map, 82 of T -graph, 305 Formal group law, 473 Equivariant map, 428 Abel, 476 elliptic, 475 in complex cobordism, 461, 477 linearisable, 473 in equivariant cobordism, 360 of geometric cobordisms, 477 in equivariant cohomology, 261, 432 universal, 475 in generalised cohomology theory, 483 Formality Euler formula, 15 in model category, 443 Euler characteristic, 254, 377, 418, 419, 483 integral, 316 Eulerian poset, 127 of dg-algebra, 411 , 397 of space 168, 315, 320, 425 of fibration, 415 F -polynomial (of polytope), 14 of pair, 420 Fr¨olicher spectral sequence, 228 Excision, 420 Full subcomplex (of simplicial complex), 63 514 INDEX

Fundamental group, 413 Gysin–Thom isomorphism, 369, 463 Fundamental homology class, 156 f-vector (face vector) Half- (left, right), 322 of cubical complex, 84 Hamiltonian action, 195, 293 of polytope, 14 Hamiltonian-minimal submanifold, 231 of simplicial complex, 60 Hamiltonian vector field, 231 of simplicial poset, 117 Hard Lefschetz Theorem, 24, 187 Hauptvermutung, 76 Gal Conjecture, 39, 74 Heisenberg group, 426 Gale diagram, 11, 208, 236 HEP (homotopy extension property), 416 combinatorial, 13, 207 Hermitian quadric, 197, 202 Gale duality, 10, 204, 210, 212, 227 Hilton–Milnor Theorem, 138 Gale transform, 10 Hirzebruch genus, 364, 480 Ganea’s Theorem, 143 equivariant, 365 g-conjecture, 73, 113 fibre multiplicative, 366 Generalised (co)homology theory, 454, oriented, 486 complex oriented, 483 rigid, 364, 366, 385 multiplicative, 457 universal, 482 Generalised Lower Bound Conjecture Hirzebruch , 182 (GLBC), 22, 188 H-minimal submanifold, 231 Generating series Hodge algebra, 116 of face polynomials, 51 Hodge number, 225 of polytopes, 49 Homological dimension (of module), 399 Genus, 364, 479 Homology equivariant, 365 cellular, 421 fibre multiplicative, 367 of chain complex, 397 oriented, 486 simplicial, 418 rigid, 366 reduced, 418 universal, 482 singular, 419 Geometric cobordism, 461 of pair, 420 Geometric quotient, 189 reduced, 419 Geometric realisation Homology polytope, 264 of cubical complex, 84 Homology sphere, 76 of simplicial complex, 59 Homotopy, 413 of simplicial set, 442 Homotopy category (of model category), Ghost vertex, 59 441 GKM-graph, 303 Homotopy cofibre, 417 GKM-manifold, 303 Homotopy colimit, 314, 450 Golod (ring, simplicial complex), 171, 345 Homotopy equivalence, 413 Gorenstein, Gorenstein* Homotopy fibre, 416 simplicial complex, 112, 154 , 413 simplicial poset, 125 Homotopy Lie algebra, 342, 423, 443 Goresky–MacPherson formula, 161 Homotopy limit, 314, 450 Graded Lie algebra, 447 Homotopy quotient, 429 Graph Homotopy type, 413 chordal, 345 Hopf Conjecture, 74 of polytope, 9 Hopf equation, 50 simple, 9, 34 Hopf line bundle, 430 Graph-associahedron, 34 Hopf manifold, 222 Graph product, 341 H-polynomial (of polytope), 14 , 188, 382 hsop (homogeneous system of parameters), g-theorem, 23, 187 408 g-vector in face rings, 105, 117 of polytope, 14 Hurewicz homomorphism, 424, 467 of simplicial complex, 60 in complex cobordism, 466 Gysin homomorphism Hurwitz series, 474, 489 in equivariant cohomology, 261, 432 h-vector in cobordism, 361, 460 of polytope, 14 INDEX 515

of simplicial complex, 60 Mayer–Vietoris sequence, 420 of simplicial poset, 117 Maximal action (of torus), 223 Hyperplane cut, 5 Meet (greatest common lower bound), 82, 114 Intersection homology, 24, 187 Milnor hypersurface, 348, 469 Intersection of quadrics, 197, 202 Minimal basis (of graded module), 400 nondegenerate (transverse), 202 Minimal model Intersection poset (of arrangement), 161 of dg-algebra, 410, 443 Isotropy representation, 381, 430 of space, 425 Minimal submanifold, 231 Join (least common upper bound), 82, 114 Minkowski sum, 26 Join (operation) Missing face, 65, 92, 332, 340 of simplicial complexes, 61 Model (of dg-algebra), 410 of simplicial posets, 172 Model category, 440 of spaces, 322 Module, 395 finitely generated, 395 Klein bottle, 234 free, 396 Koszul algebra, 403 graded, 395 Koszul complex, 403 projective, 396 Koszul resolution, 400 Moment-angle complex, 131, 172 Krichever genus, 389, 489 real, 134, 149 universal, 389, 492 Moment-angle manifold, 134, 197 K-theory, 485 polytopal, 134, 206 Lagrangian immersion, 231 non-formal, 169 Lagrangian submanifold, 231 Moment map, 195, 209 Landweber Exact Functor Theorem, 484 proper, 196, 210 Latching functor, 444 Monoid, 136 Lattice, 180, 210 Monomial ideal, 92 Lefschetz pair, 134 Moore loops, 331 Left lifting property, 416 Morava K-theory, 473 Leibniz identity, 398 Multi-fan, 239, 380 L-genus (signature), 483 Multidegree, 396 Limit (of diagram), 439 Multigrading, 100, 148, 173 Link (in simplicial complex), 62, 121 M-vector, 23 Link (of intersection of quadrics), 207 Nerve complex, 60, 191 Linkage, 130 Nested set, 31 Localisation formula, 368 Nestohedron, 31 Locally standard (torus action), 240, 257 Nilpotent space, 424 Logarithm (of formal group law), 474 Non-PL sphere, 73, 76 Loop functor (algebraic), 448 Normal complex structure, 433 Loop space, 331, 415 Lower Bound Theorem (LBT), 22 Octahedron, 4 lsop (linear system of parameters), 408 Omniorientation, 247, 261, 279, 309 in face rings, 105, 117 Opposite category, 439 integral, 106 Orbifold, 183 LVM-manifold, 213, 223 Orbit (of group action), 428 Orbit space, 429 Manifold with corners, 133, 241 Order complex (of poset), 65 face-acyclic, 264 Overcategory, 440 nice, 241 Manifold with faces, 241 Partition, 465 Mapping cone, 417 Path space, 415 Mapping cylinder, 417 Pair (of spaces), 413 , 168, 411 Perfect elimination order, 345 indecomposable, 171 Permutahedron, 28 indeterminacy of, 412 Picard group, 195 trivial (vanishing), 412 PL map, 61 Matching functor, 444 homeomorphism, 61 516 INDEX

PL manifold, 76 Projective dimension (of module), 399 PL sphere, 69 Projectivisation (of ), 463 Poincar´e algebra, 112 , 152, 307 Poincar´e–Atiyah duality, 457, 460 orientable, 153 Poincar´e duality, 421 Pullback, 439 Poincar´e duality space, 154 Pushout, 439 Poincar´e series 341, 399 of face ring, 94 Quadratic algebra, 92, 340 Poincar´e sphere, 76 Quasi-isomorphism, 398, 410, 442 Pointed map, 413 Quasitoric manifold, 244, 318, 376 Pointed space, 413 almost complex structure on, 254 Polar set, 3 canonical smooth structure on, 249 Polyhedral complex, 70 complex structure on, 254 Polyhedral product, 135, 172, 313 equivalence of, 246, 247 Polyhedral smash product, 326 Quillen pair (of functors), 441 Polyhedron (convex), 1 Quotient space, 429 Delzant, 210 Rank (of free module), 396 Polyhedron (simplicial complex), 59 Rank function, 81 Polytopal sphere, 72 Rational equivalence, 424 Polytope Rational homotopy type, 424 combinatorial, 2 Ray’s basis, 363 convex, 1 Redundant inequality, 2 cyclic, 7 Reedy category, 444, 450 Delzant, 57, 184, 199 Regular sequence, 405 dual, 4 Regular subdivision, 121 generic, 3 Regular value (of moment map), 196, 210 lattice, 183 Reisner Theorem, 107 neighbourly, 7, 17 Resolution (of module) nonrational, 185 free, 399 polar, 3 minimal, 400 regular, 4 projective, 399 self-dual, 4 Restriction (of a building set), 26 simple, 3, 46 Restriction map simplicial, 3, 46 algebraic, 94, 105, 116, 266 stacked, 22 in equivariant cohomology, 259 triangle-free, 43 Right-angled Artin group, 341 Polytope algebra, 24 Right-angled Coxeter group, 341 Pontryagin algebra, 331 Right lifting property, 416 Pontryagin class, 486 Rigidity equation, 386 Pontryagin number, 464 Ring of polytopes, 44 Pontryagin product, 423 Root (of representation), 381 Pontryagin–Thom map, 432, 454 complementary, 381 Poset (partially ordered set), 2 of almost complex structure, 381 Poset category, 439 Positive orthant, 3, 9, 131 Samelson product, 332, 423, 443 Presentation (of a polytope by higher, 333 inequalities), 2 Schlegel diagram, 70 generic, 3 Segre embedding, 469 irredundant, 2 Seifert fibration, 225 rational, 210 Sign (of fixed point), 251, 377, 434 Primitive (lattice vector), 56 Signature, 377, 483 Principal bundle, 429 Simplex, 2, 82 Principal minor (of matrix), 291 abstract, 59 Product regular, 2 of building sets, 47 standard, 2 of polytopes, 5 Simplicial cell complex, 82 Product (categorical), 439 Simplicial chain, 417 Products (in cobordism), 459 Simplicial cochain, 418 INDEX 517

Simplicial complex Syzygy, 399 abstract, 59 Alexander dual, 66 along the fibres, 250, 431 geometric, 59 Tangential representation, 430 Golod, 345 Tautological line bundle, 430 neighbourly, 141 over projectivisation, 463 pure, 59 Tensor product (of modules), 396 shifted, 171, 325 T -graph (torus graph), 304 Simplicial manifold, 76 Thom class, 267, 305, 432 Simplicial map, 60 Thom isomorphism, 467 isomorphism, 60 Thom space, 454 nondegenerate, 60 T -manifold, 239 Simplicial object (in category), 439 Todd genus (td), 378, 483 Simplicial poset, 81 Topological fan, 280 dual (of manifold with corners), 265 complete, 280 Simplicial set, 439 Topological monoid, 442 Simplicial sphere, 69 Topological toric manifold, 278 Simplicial subdivistion, 61 Tor (functor), 401 Simplicial wedge, 165 Tor-algebra, 97 Singular chain, 419 Toral rank, 162 of pair, 420 Toric manifold, 185, 237, 242 Singular cochain, 420 Hamiltonian, 199, 212 Skeleton (of cell complex), 414 over cube, 288 Slice Theorem, 430 projective, 185 Small category, 439 Toric space, xi, 130 Small cover, 234, 283, 301 Toric variety, 24, 179 Smash product, 322, 413 affine, 180 Special unitary (SU-) manifold, 389 projective, 183 Stabiliser (of group action), 428 real, 234, 283 Stably complex structure, 250, 433 Torus T -invariant, 434 algebraic, 179 Stanley–Reisner ideal, 92 standard, 130 Star, 62, 121, 218 Torus manifold, 257 Starshaped sphere, 72, 221 Triangulated manifold, 76 Stasheff polytope, 25 Triangulated sphere, 21, 69 Stationary subgroup, 428 flag, 74 Steinitz Problem, 75 Triangulation, 55, 61 Steinitz Theorem, 71, 72 neighbourly, 75, 141 Stellahedron, 37 Tropical geometry, 283 Stellar subdivision Truncated cube, 40 of simplicial complex, 78 Truncated simplex, 38 of simplicial poset, 122 Truncation, 5 Stiefel–Whitney class, 464 Type (of S1-action), 389 in equivariant cohomology, 432 Stiefel–Whitney number, 464 Undercategory, 440 Straightening relation, 116 Underlying simplicial complex (of fan), 61 Subcomplex of simplicial complex, 59 Unipotent (upper triangular matrix), 291 Sullivan algebra (of piecewise polynomial Universal G-space (universal bundle), 429 differential forms), 424 Universal toric genus, 357, 360 Supporting hyperplane, 1 Upper Bound Theorem (UBT), 20 Suspension of module, 396 Vector bundle, 429 of simplicial complex, 61 Vertex of space, 417 of polytope, 2 Suspension isomorphism, 421 of simplicial complex, 59 Symplectic manifold, 195 of simplicial poset, 81 Symplectic reduction, 196, 209, 237 Vertex truncation (vt), 5 Symplectic quotient, 196, 209 Volume polynomial, 24 518 INDEX

Weak equivalence of dg-algebras, 410 in model category, 440 Wedge (of spaces), 138, 413 of spheres, 171, 321, 345 Weierstrass ℘-function, 387, 487 Weierstrass σ-function, 487, 488 Weight (of torus representation), 250, 377, 434 Weight graph, 302 Weight lattice (of torus), 210, 434 Weyl group, 380 Whitehead product, 422 higher, 332 Witten genus, 487

Yoneda algebra, 448

Zigzag (of maps), 410 Zonotope, 29

γ-polynomial, 18 γ-vector of polytope, 18 of triangulated sphere, 74 χa,b-genus, 373, 483 χy-genus, 373, 483

2-truncated cube, 40 24-cell, 4 5-lemma, 398 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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