Evaluation maps and transfers for free loop spaces I

TL;DR

This paper develops a functorial extension of the evaluation map for free loop spaces, incorporating transfers along finite covers, with algebraic formulas for classifying spaces of finite groups, impacting the homotopy category of spectra.

Contribution

It introduces a new functorial extension of the evaluation map for free loop spaces, with algebraic formulas for finite group classifying spaces, advancing the understanding of transfers in algebraic topology.

Findings

Constructed a functorial extension of the evaluation map.

Provided algebraic formulas for classifying spaces of finite groups.

Established a natural evaluation map on p-completed classifying spectra.

Abstract

We construct and study a functorial extension of the evaluation map $S^1 \times \mathcal{L} X \to X$ to transfers along finite covers. For finite covers of classifying spaces of finite groups, we provide algebraic formulas for this extension in terms of bisets. In the sequel, 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.