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Torus Actions and Their Applications in Topology and Combinatorics
Victor M. Buchstaber and Taras E. Panov, Moscow State University, Russia
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University Lecture Series
2002; 144 pp; softcover
Volume: 24
ISBN-10: 0-8218-3186-0
ISBN-13: 978-0-8218-3186-1
List Price: US$34
Member Price: US$27.20
Order Code: ULECT/24
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See also:

Toric Varieties - David A Cox, John B Little and Henry K Schenck

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.

Readership

Graduate students and research mathematicians interested in topology or combinatorics; topologists interested in combinatorial applications and vice versa.

Reviews

"The book is quite well-written and includes many new and well-known open problems"

-- Mathematical Reviews

"The text contains a wealth of material and ... the book may be a welcome collection for researchers in the field and a useful overview of the literature for novices."

-- Zentralblatt MATH

Table of Contents

  • Introduction
  • Polytopes
  • Topology and combinatorics of simplicial complexes
  • Commutative and homological algebra of simplicial complexes
  • Cubical complexes
  • Toric and quasitoric manifolds
  • Moment-angle complexes
  • Cohomology of moment-angle complexes and combinatorics of triangulated manifolds
  • Cohomology rings of subspace arrangement complements
  • Bibliography
  • Index
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