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The $S^1$ assembly map on $K\mkern-2mu$-theory and topological cyclic homology

Noah Riggenbach
Vol. 10 (2025), No. 3, 447–471
DOI: 10.2140/akt.2025.10.447
Abstract

We study the assembly map on K-theory and on topological cyclic homology, written as TC ⁡ . Specifically, we study the maps

S+1 ∧ K(𝕊) → K(𝕊[x±1]) andS +1 ∧ TC ⁡ (𝕊) → TC ⁡ (𝕊[x±1]).

In the second case we are able to describe geometrically what the cofiber is. Using this we show that the Whitehead space of the circle has a countable sum of Ω∞coker ⁡ (j)p∧ as a summand. This paper was part of the author’s thesis.

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Keywords
$K\mkern-2mu$-theory, topological cyclic homology, trace methods, Whitehead spectrum
Mathematical Subject Classification
Primary: 19D10
Secondary: 19D55, 55P91
Milestones
Received: 4 February 2023
Revised: 17 May 2024
Accepted: 10 June 2025
Published: 21 July 2025
Authors
Noah Riggenbach
Department of Mathematics
Northwestern University
Evanston, IL
United States
© 2025 MSP (Mathematical Sciences Publishers).