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#### Higher topological Hochschild homology of periodic complex K-theory

##### External Resource

https://doi.org/10.1016/j.topol.2020.107302

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##### Fulltext (public)

arXiv:1801.00156.pdf

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##### Citation

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.

Cite as: https://hdl.handle.net/21.11116/0000-0006-E4FA-F

##### 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 $KU$.

$K$-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 $KU$.