The transfer map of free loop spaces

TL;DR

This paper develops a transfer map for free loop spaces using topological Hochschild homology, demonstrating its compatibility with existing transfer maps and providing new geometric models and computations.

Contribution

It introduces a new free loop transfer map compatible with the Becker-Gottlieb transfer and extends known results in $A$-theory and free loop space computations.

Findings

Established compatibility with Becker-Gottlieb transfer

Provided a geometric model via Pontryagin-Thom collapse maps

Reproduced and extended previous free loop transfer computations

Abstract

For any perfect fibration $E \rightarrow B$, there is a "free loop transfer map" $LB_+ \rightarrow LE_+$, defined using topological Hochschild homology. We prove that this transfer is compatible with the Becker-Gottlieb transfer, allowing us to extend a result of Dorabia\l{}a and Johnson on the transfer map in Waldhausen's $A$-theory. In the case where $E \rightarrow B$ is a smooth fiber bundle, we also give a concrete geometric model for the free loop transfer in terms of Pontryagin-Thom collapse maps. We recover the previously known computations of the free loop transfer due to Schlichtkrull, and make a few new computations as well.