The transfer map of free loop spaces

Abstract

For any perfect fibration $E\to B$, there is a "free loop transfer map" $L{B}_{+}\to L{E}_{+}$, defined using topological Hochschild homology. We prove that this transfer is compatible with the Becker-Gottlieb transfer, allowing us to extend a result of Dorabiała and Johnson on the transfer map in Waldhausen's $A$-theory. In the case where $E\to B$ 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.

arXiv commentv. 2; new string diagrams added, 54 pages, 7 figures. To appear in Trans. AMS