About this Title
Victor M. Buchstaber, Steklov Mathematical Institute, Moscow, Russia and Taras E. Panov, Moscow State University, Russia
Publication: Mathematical Surveys and Monographs
Publication Year: 2015; Volume 204
ISBNs: 978-1-4704-2214-1 (print); 978-1-4704-2499-2 (online)
MathSciNet review: MR3363157
MSC: Primary 57Sxx; Secondary 13F55, 14M25, 52B20, 55N22, 55P62, 57Qxx
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.
Graduate students and research mathematicians interested in algebraic topology.
Table of Contents
- Chapter 1. Geometry and combinatorics of polytopes
- Chapter 2. Combinatorial structures
- Chapter 3. Combinatorial algebra of face rings
- Chapter 4. Moment-angle complexes
- Chapter 5. Toric varieties and manifolds
- Chapter 6. Geometric structures on moment-angle manifolds
- Chapter 7. Half-dimensional torus actions
- Chapter 8. Homotopy theory of polyhedral products
- Chapter 9. Torus actions and complex cobordism
- Appendix A. Commutative and homological algebra
- Appendix B. Algebraic topology
- Appendix C. Categorical constructions
- Appendix D. Bordism and cobordism
- Appendix E. Formal group laws and Hirzebruch genera