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Article Dans Une Revue Topology and its Applications Année : 2020

Higher topological Hochschild homology of periodic complex K-theory

Résumé

We describe the topological Hochschild homology of the periodic complex $K$-theory spectrum, $THH(KU)$, as a commutative $KU$-algebra: it is equivalent to $KU[K(\Z,3)]$ and to $F(\Sigma KU_\Q)$, where $F$ is the free commutative $KU$-algebra functor on a $KU$-module. Moreover, $F(\Sigma KU_\Q)\simeq KU \vee \Sigma KU_\Q$, 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 $KU[K(\Z,n+2)]$ and to $F(\Sigma^n KU_\Q)$. We describe the topological André-Quillen homology of $KU$ as $KU_\Q$.
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Dates et versions

hal-04007385 , version 1 (28-02-2023)

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Bruno Stonek. Higher topological Hochschild homology of periodic complex K-theory. Topology and its Applications, 2020, 282 (107302), ⟨10.1016/j.topol.2020.107302⟩. ⟨hal-04007385⟩
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