Evaluation maps and transfers for free loop spaces II

TL;DR

This paper extends the evaluation map for free loop spaces to a broader context involving classifying spectra of finite groups, revealing new functorial properties and connections with fusion systems.

Contribution

It introduces a natural evaluation map on p-completed classifying spectra, linking free loop space transfers with the Burnside category of fusion systems.

Findings

Established a functorial evaluation map on classifying spectra.

Connected free loop space transfers with fusion system Burnside categories.

Extended previous work to a broader class of spectra.

Abstract

In our previous paper, we constructed and studied a functorial extension of the evaluation map $S^1 \times \mathcal{L}X \to X$ to transfers along finite covers. In this paper, we show that this induces a natural evaluation map on the full subcategory of the homotopy category of spectra consisting of $p$-completed classifying spectra of finite groups. To do this, we leverage the close relationship between this full subcategory and the Burnside category of fusion systems.