The transfer map of free loop spaces

Lind, John; Malkiewich, Cary

doi:10.1090/tran/7497

The transfer map of free loop spaces

Authors: John A. Lind and Cary Malkiewich
Journal: Trans. Amer. Math. Soc. 371 (2019), 2503-2552
MSC (2010): Primary 55R12; Secondary 19D10, 19D55, 55J35
DOI: https://doi.org/10.1090/tran/7497
Published electronically: November 27, 2018
MathSciNet review: 3896088
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Abstract: For any perfect fibration $E \longrightarrow B$, there is a “free loop transfer map” $LB_+ \longrightarrow LE_+$, defined using topological Hochschild homology. We prove that this transfer is compatible with the Becker-Gottlieb transfer, allowing us to extend a result of Dorabiała and Johnson on the transfer map in Waldhausen’s $A$-theory. In the case where $E \longrightarrow B$ 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.

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