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Toric Topology Mathematical Surveys and Monographs Volume 204 Toric Topology 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 Mathematics 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. Algebraic topology. 4. Geometry, Algebraic. I. Panov, Taras E., 1975– II. Title QA613.2.B82 2015 516.35–dc23 2015006771 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 Contents Introduction ix Chapter guide x Acknowledgements xiii Chapter 1. Geometry and Combinatorics of Polytopes 1 1.1. Convex polytopes 1 1.2. Gale duality 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. Alexander duality 66 2.5. Classes of triangulated spheres 69 2.6. Triangulated manifolds 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. Cohomology 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 Dolbeault cohomology 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. Homotopy Theory of Polyhedral Products 313 8.1. Rational homotopy theory 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 Cobordism 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 Homological Algebra 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 homology 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 cobordisms 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 mathematical physics, 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 algebraic geometry, 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 manifold 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.
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