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.

Foreword

vii

I Notes on String Topology

1

(96)

Ralph L. Cohen

Alexander A. Voronov

Introduction

3

(2)

Intersection theory in loop spaces

5

(20)

Intersections in compact manifolds

6

(2)

The Chas-Sullivan loop product

8

(2)

The BV structure and the string bracket

10

(5)

A stable homotopy point of view

15

(4)

Relation to Hochschild cohomology

19

(6)

The cacti operad

25

(22)

PROPs and operads

25

(13)

PROP's

25

(1)

Algebras over a PROP

26

(1)

Operads

27

(1)

Algebras over an operad

28

(8)

Operads via generators and relations

36

(2)

The cacti operad

38

(5)

The cacti action on the loop space

43

(4)

Action via correspondences

43

(2)

The BV structure

45

(2)

String topology as field theory

47

(20)

Field theories

47

(6)

Topological Field Theories

47

(4)

(Topological) Conformal Field Theories

51

(1)

Examples

51

(2)

Motivic TCFTs

53

(1)

Generalized string topology operations

53

(8)

Open-closed string topology

61

(6)

A Morse theoretic viewpoint

67

(12)

Cylindrical gradient graph flows

67

(5)

Cylindrical holomorphic curves in T*M

72

(7)

Brane topology

79

(18)

The higher-dimensional cacti operad

79

(2)

The cacti action on the sphere space

81

(1)

The algebraic structure on homology

82

(2)

Sphere spaces and Hochschild homology

84

(5)

Bibliography

89

(8)

II An Algebraic Model for Mod 2 Topological Cyclic Homology

97

(64)

Kathryn Hess

Preface

99

(4)

Preliminaries

103

(12)

Elementary definitions, terminology and notation

103

(3)

The canonical, enriched Adams-Hilton model

106

(3)

Twisting cochains

106

(1)

Strongly homotopy coalgebra and comodule maps

106

(2)

The canonical Adams-Hilton model

108

(1)

Noncommutative algebraic models of fiber squares

109

(6)

Free loop spaces

115

(24)

A simplicial model for the free loop space

115

(6)

The general model

115

(3)

Choosing the free loop model functorially

118

(3)

The multiplicative free loop space model

121

(5)

The diagonal map

121

(1)

The path fibration

122

(2)

The free loop space model

124

(2)

The free loop model for topological spaces

126

(7)

Linearization of the free loop model

133

(6)

Homotopy orbit spaces

139

(14)

A special family of primitives

139

(2)

A useful resolution of CU*ES1

141

(1)

Modeling S1-homotopy orbits

142

(2)

The case of the free loop space

144

(9)

A model for mod 2 topological cyclic homology

153

(8)

The pth- power map

153

(4)

Topological cyclic homology

157

(4)

Bibliography

161

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String Topology And Cyclic Homology

9783764321826

3764321822

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