The string coproduct “knows” Reidemeister/Whitehead torsion

Naef, Florian

doi:10.2140/gt.2024.28.3643

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The string coproduct “knows” Reidemeister/Whitehead torsion

Florian Naef

Geometry & Topology 28 (2024) 3643–3659

DOI: 10.2140/gt.2024.28.3643

Abstract

We show that the string coproduct is not homotopy invariant. More precisely, we show that the (reduced) coproducts are different on $L (1, 7)$ and $L (2, 7)$ . Moreover, the coproduct on $L (k, 7)$ can be expressed in terms of the Reidemeister torsion and hence transforms with respect to the Whitehead torsion of a homotopy equivalence. The string coproduct can thereby be used to compute the image of the Whitehead torsion under the Dennis trace map.

Keywords

string topology, lens spaces, Reidemeister torsion, Whitehead torsion

Mathematical Subject Classification

Primary: 55P50

References

Publication

Received: 4 July 2021

Revised: 14 November 2022

Accepted: 16 April 2023

Published: 20 December 2024

Proposed: Jesper Grodal

Seconded: Ciprian Manolescu, Stefan Schwede

Authors

Florian Naef
Department of Mathematical Sciences University of Copenhagen Copenhagen Denmark	School of Mathematics Trinity College Dublin Dublin Ireland

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Volume 28, issue 8 (2024)