A Chen model for mapping spaces and the surface product

TL;DR

This paper introduces a new algebraic framework using Chen iterated integrals to model mapping spaces and define a surface product operation, extending string topology concepts to higher genus surfaces with applications to spheres and Lie groups.

Contribution

It develops a novel machinery of Chen iterated integrals for higher Hochschild complexes, providing algebraic models for mapping spaces and defining a homotopy-invariant surface product.

Findings

Surface product is homotopy invariant.

Explicit formulas for surface product on odd spheres.

Hochschild-Kostant-Rosenberg type theorems established.

Abstract

We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups.