Higher operations in string topology of classifying spaces

TL;DR

This paper computes numerous non-trivial higher string topology operations for classifying spaces of various Lie groups, revealing new homology classes in related automorphism and holomorph groups.

Contribution

It provides explicit calculations of higher string topology operations in the context of classifying spaces, a rare achievement in the literature.

Findings

Discovery of many non-trivial higher string topology operations.

Identification of new homology classes in automorphism groups of free groups.

Results applicable to affine groups over integers and finite fields.

Abstract

Examples of non-trivial higher string topology operations have been regrettably rare in the literature. In this paper, working in the context of string topology of classifying spaces, we provide explicit calculations of a wealth of non-trivial higher string topology operations associated to a number of different Lie groups. As an application of these calculations, we obtain an abundance of interesting homology classes in the twisted homology groups of automorphism groups of free groups, the ordinary homology groups of holomorphs of free groups, and the ordinary homology groups of affine groups over the integers and the field of two elements.