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  Higher topological Hochschild homology of periodic complex K-theory

Stonek, B. (2020). Higher topological Hochschild homology of periodic complex K-theory. Topology and its Applications, 282: 107302. doi:10.1016/j.topol.2020.107302.

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https://doi.org/10.1016/j.topol.2020.107302 (Publisher version)
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 Creators:
Stonek, Bruno1, Author           
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Algebraic Topology
 Abstract: We describe the topological Hochschild homology of the periodic complex
$K$-theory spectrum, $THH(KU)$, as a commutative $KU$-algebra: it is equivalent
to $KU[K(\mathbb{Z},3)]$ and to $F(\Sigma KU_{\mathbb{Q}})$, where $F$ is the
free commutative $KU$-algebra functor on a $KU$-module. Moreover, $F(\Sigma
KU_{\mathbb{Q}})\simeq KU \vee \Sigma KU_{\mathbb{Q}}$, a square-zero
extension. In order to prove these results, we first establish that topological
Hochschild homology commutes, as an algebra, with localization at an element.
Then, we prove that $THH^n(KU)$, the $n$-fold iteration of $THH(KU)$, i.e.
$T^n\otimes KU$, is equivalent to $KU[G]$ where $G$ is a certain product of
integral Eilenberg-Mac Lane spaces, and to a free commutative $KU$-algebra on a
rational $KU$-module. We prove that $S^n \otimes KU$ is equivalent to
$KU[K(\mathbb{Z},n+2)]$ and to $F(\Sigma^n KU_{\mathbb{Q}})$. We describe the
topological Andr\'e-Quillen homology of $KU$.

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Language(s): eng - English
 Dates: 2020
 Publication Status: Published in print
 Pages: 43
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Degree: -

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Title: Topology and its Applications
  Abbreviation : Topology Appl.
Source Genre: Journal
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Publ. Info: Elsevier
Pages: - Volume / Issue: 282 Sequence Number: 107302 Start / End Page: - Identifier: -