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Mathematical Surveys
and Monographs
Volume 204
American Mathematical Society
Toric Topology
Victor M. Buchstaber Taras E. Panov
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
QA613.2.B82 2015516.3′5–dc23 2015006771
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10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15
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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