String topology prospectra and Hochschild cohomology

TL;DR

This paper explores the relationship between string topology for classifying spaces of compact Lie groups and Hochschild cohomology, establishing equivalences and dualities between various spectra and algebraic structures.

Contribution

It identifies the string topology prospectrum with homotopy fixed points and orbits, linking these to Hochschild cohomology and Koszul duality, revealing new algebraic structures.

Findings

Equivalence of string topology prospectrum with homotopy fixed-point spectrum.

Identification of dual spectra with Hochschild cohomology products.

Connection between homology products and Gerstenhaber cup product.

Abstract

We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group $G$, we show that the string topology prospectrum $LBG^{-TBG}$ is equivalent to the homotopy fixed-point prospectrum for the conjugation action of $G$ on itself, $G^{hG}$. Dually, we identify $LBG^{-ad}$ with the homotopy orbit spectrum $(DG)_{hG}$, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product in the Hochschild cohomology of $C^*(BG)$ and $C_*(G)$, respectively. These, in turn, are isomorphic via Koszul duality.