**University Lecture Series**

Volume: 24;
2002;
144 pp;
Softcover

MSC: Primary 52; 57; 14; 13;

Print ISBN: 978-0-8218-3186-1

Product Code: ULECT/24

List Price: $36.00

AMS Member Price: $28.80

MAA Member Price: $32.40

**Electronic ISBN: 978-1-4704-2171-7
Product Code: ULECT/24.E**

List Price: $36.00

AMS Member Price: $28.80

MAA Member Price: $32.40

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# Torus Actions and Their Applications in Topology and Combinatorics

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*Victor M. Buchstaber; Taras E. Panov*

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 & Endorsements

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

# Table of Contents

## Torus Actions and Their Applications in Topology and Combinatorics

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction 110 free
- Chapter 1. Polytopes 716 free
- Chapter 2. Topology and combinatorics of simplicial complexes 2130
- Chapter 3. Commutative and homological algebra of simplicial complexes 3544
- Chapter 4. Cubical complexes 4958
- Chapter 5. Toric and quasitoric manifolds 5766
- Chapter 6. Moment-angle complexes 8594
- 6.1. Moment-angle manifolds Z[sub(p)] defined by simple polytopes 8594
- 6.2. General moment-angle complexes Z[sub(K)] 8796
- 6.3. Cell decompositions of moment-angle complexes 8998
- 6.4. Moment-angle complexes corresponding to joins, connected sums and bistellar moves 92101
- 6.5. Borel constructions and Davis-Januszkiewicz space 94103
- 6.6. Walk around the construction of Z[sub(K)]: generalizations, analogues and additional comments 97106

- Chapter 7. Cohomology of moment-angle complexes and combinatorics of triangulated manifolds 101110
- 7.1. The Eilenberg-Moore spectral sequence 101110
- 7.2. Cohomology algebra of Z[sub(K)] 102111
- 7.3. Bigraded Betti numbers of Z[sub(K): the case of general K 106115
- 7.4. Bigraded Betti numbers of Z[sub(K)]: the case of spherical K 110119
- 7.5. Partial quotients of Z[sub(p)] 113122
- 7.6. Bigraded Poincaré duality and Dehn-Sommerville equations 117126

- Chapter 8. Cohomology rings of subspace arrangement complements 125134
- Bibliography 135144
- Index 141150
- Back Cover Back Cover1154