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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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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Additional Information

John A. Lind
Affiliation: Department of Mathematics, Reed College, Portland, Oregon 97202

Cary Malkiewich
Affiliation: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902-6000

DOI: https://doi.org/10.1090/tran/7497
Received by editor(s): May 19, 2016
Received by editor(s) in revised form: August 27, 2017
Published electronically: November 27, 2018
Additional Notes: The first author was partially supported by the DFG through SFB1085
The second author was partially supported by an AMS Simons Travel Grant
Article copyright: © Copyright 2018 American Mathematical Society