Rational String Topology

TL;DR

This paper leverages rational homotopy theory to explicitly model string topology operations on 1-connected closed manifolds, establishing isomorphisms with Hochschild cohomology and providing concrete computations.

Contribution

It introduces an explicit cochain model for string topology operations using rational homotopy theory, connecting loop homology to Hochschild cohomology.

Findings

Loop homology is isomorphic to Hochschild cohomology of A_{PL}(M).

Explicit computations of the loop product and string bracket are provided.

The model simplifies calculations of string topology invariants.

Abstract

We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a 1-connected closed manifold M. We prove that the loop homology of M is isomorphic to the Hochschild cohomology of the commutative graded algebra A_{PL}(M) with coefficients in itself. Some explicit computations of the loop product and the string bracket are given.