The transfer map of free loop spaces

Abstract

Abstract. For any perfect fibration E −→ B, there is a "free loop transfer map" LB + −→ 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 la and Johnson on the transfer map in Waldhausen's A-theory. In the case where E −→ 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.

The transfer is functorial

“…The free loop transfer is functorial up to homotopy, using for instance [LM,Thm 1.10] and [PS,Prop 7.5]. Chasing diagrams, we conclude that the commutativity of the above square for all such fibrations p implies Theorem A.…”

“…We will be content to discuss a possible new approach to Bass's strong trace conjecture [B], which would not be able to get off the ground were Theorem A not true in its fullest generality. We will only briefly indicate the idea -more thorough references for the concepts here can be found in [G2,LM].…”

Topological Hochschild homology and higher characteristics

“…Remark 6.4. The above was proved in [LM16], as a step in verifying the first-named author's conjecture that the Becker-Gottlieb transfer factors through the THH transfer.…”

On transfer maps in the algebraic K-theory of spaces

“…Assuming that B is homotopy finite, this claim is then a special case of Theorem 3.7. For the proof of Theorem 3.7, we consider a relative version of the A-theory characteristic of Dwyer-Weiss-Williams and use the theorem of Lind-Malkiewich [9] (stated in (A)) in order to express the composition…”