Batalin–Vilkovisky coalgebra of string topology

Batalin-vilkovisky Coalgebra of String Topology

We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gr...

String topology for stacks

We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. In particular, we give a good notion of a free loop stack, and of a mapping stack Map(Y,X), where Y is a compact space and X a topological stack, which is functorial both in X and Y and behaves well enough with respect to p...

Higher String Topology Operations

In [1] Chas and Sullivan have defined an intersection-type product on the homology of the free loop space LM of an oriented manifold M. In this paper we show how to extend this construction to a topological conformal field theory of degree d. In particular, we get operations on H∗LM which are parameterized by the homology of the moduli space of open-closed Riemann surfaces.

Notes on String Topology

String Topology and Cyclic Homology

This book is devoted to some recent developments in our understanding of the topology of free loop spaces of manifolds. The contributions are based on three lecture courses given by the authors at a summer school at Almeŕıa, Spain, in the summer of 2003. The two parts treat rather different facets of the theory, and I will try to give some perspective on each of them in turn. The free loop spac...