Torus Actions and Their Applications in Topology and Combinatorics
About this Title
Victor M. Buchstaber, Moscow State University, Moscow, Russia and Taras E. Panov, Moscow State University, Moscow, Russia
Publication: University Lecture Series
Publication Year 2002: Volume 24
ISBNs: 978-0-8218-3186-1 (print); 978-1-4704-2171-7 (online)
MathSciNet review: MR1897064
MSC: Primary 57Q15; Secondary 13F55, 14M25, 52B70, 57R19
The book presents the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This established link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits. Conversely, subtle properties of a combinatorial object can be realized by interpreting it as the orbit structure for a proper manifold or as a complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc., while the combinatorial objects include simplicial and cubical complexes, polytopes, and arrangements. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics.
The exposition centers around the theory of moment-angle complexes, providing an effective way to study invariants of triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook. It will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the subjects involved.
Graduate students and research mathematicians interested in topology or combinatorics; topologists interested in combinatorial applications and vice versa.
Table of Contents
- Chapter 1. Polytopes
- Chapter 2. Topology and combinatorics of simplicial complexes
- Chapter 3. Commutative and homological algebra of simplicial complexes
- Chapter 4. Cubical complexes
- Chapter 5. Toric and quasitoric manifolds
- Chapter 6. Moment-angle complexes
- Chapter 7. Cohomology of moment-angle complexes and combinatorics of triangulated manifolds
- Chapter 8. Cohomology rings of subspace arrangement complements