On string topology of classifying spaces

TL;DR

This paper extends string topology operations from classifying spaces of Lie groups to more general spaces homotopy equivalent to finite graphs, creating a new field theory related to automorphisms of free groups.

Contribution

It introduces a novel construction replacing diffeomorphisms with homotopy equivalences and surfaces with boundary with spaces homotopy equivalent to finite graphs.

Findings

Established a new string topology field theory for classifying spaces.

Connected algebraic structures in string topology to automorphism groups of free groups.

Extended known results to a broader class of spaces beyond manifolds.

Abstract

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.