Toric Topology

Front Cover
American Mathematical Soc., Jul 15, 2015 - Mathematics - 518 pages

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, combinatorics, 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 algebraic geometry of toric varieties via the notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to important connections 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.

 

 

Selected pages

Contents

Geometry and Combinatorics of Polytopes
1
Combinatorial Structures
55
Combinatorial Algebra of Face Rings
91
MomentAngle Complexes
129
Toric Varieties and Manifolds
179
Geometric Structures on MomentAngle Manifolds
201
HalfDimensional Torus Actions
239
Homotopy Theory of Polyhedral Products
313
Appendix A Commutative and Homological Algebra
395
Appendix B Algebraic Topology
413
Categorical Constructions
439
Bordism and Cobordism
453
Appendix E Formal Group Laws and Hirzebruch Genera
473
Bibliography
495
Index
511
Copyright

Torus Actions and Complex Cobordism
347

Common terms and phrases

algebra assume bordism bordism class Bott tower building set cell complex Chern class cobordism cochain cochain complex coefficients cofibrant Cohen–Macaulay cohomology ring colimits combinatorial commutative cone connected sum consider construction coordinate Corollary corresponding decomposition defined Definition denote dg-algebra diagram differential dimension dual elements equation equivariant Example face ring facets fibration fibre finite fixed points follows formal group law formula functor generalised genus geometric given graded h-vector homology homomorphism homotopy equivalent hyperplane implies induced intersection isomorphism lattice Lemma line bundle linear Massey product matrix moment-angle complex moment-angle manifold monomials n-polytope obtain orbit oriented polyhedral product polynomial poset projective PROOF Proposition quadrics quasi-isomorphism quasitoric manifold quotient result simple polytope simplex simplicial complex simplicial poset space sphere stably complex structure subcomplex subdivision subgroup subset subspace symplectic Theorem topological toric manifold toric variety torus action triangulated trivial vectors vertex vertices weak equivalence zero

About the author (2015)

Victor M. Buchstaber, Steklov Mathematical Institute, Moscow, Russia, and Taras E. Panov, Moscow State University, Russia

 

Bibliographic information

TitleToric Topology
Volume 204 of Mathematical Surveys and Monographs
AuthorsVictor M. Buchstaber, Taras E. Pano
Editionillustrated
PublisherAmerican Mathematical Soc., 2015
ISBN147042214X, 9781470422141
Length518 pages
Subjects ›  › 

Mathematics / Algebra / General
Mathematics / Geometry / Algebraic
Mathematics / Geometry / General
Mathematics / Topology
  
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