String topology and cyclic homology / Ralph L. Cohen, Kathryn Hess, Alexander A. Voronov.

Format:

Book

Author/Creator:

Cohen, Ralph L., 1952-

Contributor:

Hess, Kathryn, 1967-

Voronov, Alexander A.

Series:

Advanced courses in mathematics, CRM Barcelona

Language:

English

Subjects (All):

Loop spaces.

Homotopy theory.

Complex manifolds.

Differential topology.

Algebraic topology.

Physical Description:

vi, 163 pages : illustrations ; 24 cm.

Place of Publication:

Basel ; Boston : Birkhäuser, 2006.

Summary:

The subject of this book is string topology, Hochschild and cyclic homology. The first part consists of an excellent exposition of various approaches to string topology and the Chas-Sullivan loop product. The second gives a complete and clear construction of an algebraic model for computing topological cyclic homology. The book provides many references for the reader wishing to learn more about the subject, to which it gives a perfect introduction. It is therefore suitable for both graduate students and established researchers. It is certainly the best source of much information that was until now available only to specialists and covers material from the elementary bases to the most recent developments.

Contents:

I Notes on String Topology / Ralph L. Cohen, Alexander A. Voronov 1

1 Intersection theory in loop spaces 5

1.1 Intersections in compact manifolds 6

1.2 The Chas-Sullivan loop product 8

1.3 The BV structure and the string bracket 10

1.4 A stable homotopy point of view 15

1.5 Relation to Hochschild cohomology 19

2 The cacti operad 25

2.1 PROPs and operads 25

2.1.1 PROP's 25

2.1.2 Algebras over a PROP 26

2.1.3 Operads 27

2.1.4 Algebras over an operad 28

2.1.5 Operads via generators and relations 36

2.2 The cacti operad 38

2.3 The cacti action on the loop space 43

2.3.1 Action via correspondences 43

2.3.2 The BV structure 45

3 String topology as field theory 47

3.1 Field theories 47

3.1.1 Topological Field Theories 47

3.1.2 (Topological) Conformal Field Theories 51

3.1.4 Motivic TCFTs 53

3.2 Generalized string topology operations 53

3.3 Open-closed string topology 61

4 A Morse theoretic viewpoint 67

4.1 Cylindrical gradient graph flows 67

4.2 Cylindrical holomorphic curves in T* M 72

5 Brane topology 79

5.1 The higher-dimensional cacti operad 79

5.2 The cacti action on the sphere space 81

5.3 The algebraic structure on homology 82

5.4 Sphere spaces and Hochschild homology 84

II An Algebraic Model for Mod 2 Topological Cyclic Homology / Kathryn Hess 97

1.1 Elementary definitions, terminology and notation 103

1.2 The canonical, enriched Adams-Hilton model 106

1.2.1 Twisting cochains 106

1.2.2 Strongly homotopy coalgebra and comodule maps 106

1.2.3 The canonical Adams-Hilton model 108

1.3 Noncommutative algebraic models of fiber squares 109

2 Free loop spaces 115

2.1 A simplicial model for the free loop space 115

2.1.1 The general model 115

2.1.2 Choosing the free loop model functorially 118

2.2 The multiplicative free loop space model 121

2.2.1 The diagonal map 121

2.2.2 The path fibration 122

2.2.3 The free loop space model 124

2.3 The free loop model for topological spaces 126

2.4 Linearization of the free loop model 133

3 Homotopy orbit spaces 139

3.1 A special family of primitives 139

3.2 A useful resolution of CU[subscript *]ES[superscript 1] 141

3.3 Modeling S[superscript 1]-homotopy orbits 142

3.4 The case of the free loop space 144

4 A model for mod 2 topological cyclic homology 153

4.1 The p[superscript th]- power map 153

4.2 Topological cyclic homology 157.

Notes:

Includes bibliographical references (pages [161]-163).

ISBN:

3764321822

9783764321826

3764373881

9783764373887

OCLC:

62742673

Publisher Number:

99948069763

Online:

Publisher description