This book explores string topology, Hochschild and cyclic homology, assembling material from a wide scattering of scholarly sources in a single practical volume. The first part offers a thorough and elegant exposition of various approaches to string topology and the ChasSullivan loop product. The second gives a complete and clear construction of an algebraic model for computing topological cyclic homology.

foreword

i notes on string topology

ralph l. cohen and alexander a. voronov

introduction

1 intersection theory in loop spaces

1.1 intersections in compact manifolds

1.2 the chas-sullivan loop product

1.3 the bv structure and the string bracket

1.4 a stable homotopy point of view

1.5 relation to hochschild cohomology

2 the cacti operad

2.1 props and operads

2.1.1 prop's

2.1.2 algebras over a prop

2.1.3 operads

2.1.4 algebras over an operad

2.1.5 operads via generators and relations

2.2 the cacti operad

2.3 the cacti action on the loop space

.2.3.1 action via correspondences

2.3.2 the bv structure

3 string topology as field theory

3.1 field theories

3.1.1 topological field theories

3.1.2 (topological) conformal field theories

3.1.3 examples

3.1.4 motivic tcfts

3.2 generalized string topology operations

3.3 open-closed string topology

4 a morse theoretic viewpoint

4.1 cylindrical gradient graph flows

4.2 cylindrical holomorphic curves in t*m

5 brahe topology

5.1 the higher-dimensional cacti operad

5.2 the cacti action on the sphere space

5.3 the algebraic structure on homology

5.4 sphere spaces and hochschild homology

bibliography

ii an algebraic model for mod 2 topological cyclic homology

kathryn hess

preface

1 preliminaries

1.1 elementary definitions, terminology and notation

1.2 the canonical, enriched adams-hilton model

1.2.1 twisting cochains

1.2.2 strongly homotopy coalgebra and comodule maps

1.2.3 the canonical adams-hilton model

1.3 noncommutative algebraic models of fiber squares

2 free loop spaces

2.1 a simplicial model for the free loop space

2.1.1 the general model

2.1.2 choosing the free loop model functorially

2.2 the multiplicative free loop space model

2.2.1 the diagonal map

2.2.2 the path fibration

2.2.3 the free loop space model

2.3 the free loop model for topological spaces

2.4 linearization of the free loop model

3 homotopy orbit spaces

3.1 a special family of primitives

3.2 a useful resolution of cu, es1

3.3 modeling s1-homotopy orbits

3.4 the case of the free loop space

4 a model for mod 2 topological cyclic homology

4.1 the pth-power map

4.2 topological cyclic homology

bibliography