# Topological Hochschild homology

• ## General Introduction

A homology theory for ring spectra.

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• ## Online References

An interesting paper by Shipley

Page 33 of Weibel

Greenlees: Spectra for commutative algebraists (Homotopy theory folder). Applications: Section 6A discussion Topological HH, section 6B discusses trace maps.

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• ## Paper References

A brief and nice introduction is found in Hesselholt's chapter in the K-theory handbook.

Bokstedt: Topological Hochshild homology. Preprint, 1985.

Waldhausen et al in JLMS: THH article

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• ## Computations and Examples

Rognes: K118, K119, K120

Ausoni

[arXiv:1006.4347] Topological Hochschild Homology of $K/p$ as a $Kp^\wedge$ module from arXiv Front: math.AT by Samik Basu Let $R$ be an $E\infty$-ring spectrum. Given a map $\zeta$ from a space $X$ to $BGL1R$, one can construct a Thom spectrum, $X^\zeta$, which generalises the classical notion of Thom spectrum for spherical fibrations in the case $R=S^0$, the sphere spectrum. If $X$ is a loop space ($\simeq \Omega Y$) and $\zeta$ is homotopy equivalent to $\Omega f$ for a map $f$ from $Y$ to $B^2GL1R$, then the Thom spectrum has an $A\infty$-ring structure. The Topological Hochschild Homology of these $A\infty$-ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of $Y$.

This paper considers the case $X=S^1$, $R=Kp^\wedge$, the p-adic $K$-theory spectrum, and $\zeta = 1-p \in \pi1BGL1Kp^\wedge$. The associated Thom spectrum $(S^1)^\zeta$ is equivalent to the mod p $K$-theory spectrum $K/p$. The map $\zeta$ is homotopy equivalent to a loop map, so the Thom spectrum has an $A\infty$-ring structure. I will compute $\pi*THH^{K_p^\wedge}(K/p)$ using its description as a Thom spectrum.

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• ## History and Applications

Introduced by Bokstedt in the early 80s.

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• ## Some Research Articles

Preprint in progress of Rognes: On the Tate construction of topological Hochschild homology and its relation to the construction of Singer

Preprint in progress of Rognes: Topological Hochschild homology of topological modular forms

Localization for THH(ku) and the topological Hochschild and cyclic homology of Waldhausen categories, by Blumberg and Mandell: Link. Abstract: We prove a conjecture of Hesselholt and Ausoni-Rognes, establishing localization cofiber sequences of spectra for THH(ku) and TC(ku). These sequences support Hesselholt's view of the map l to ku as a "tamely ramified" extension of ring spectra, and validate the hypotheses necessary for Ausoni's simplified computation of V(1)_* K(KU). In order to make sense of the relative term THH(ku|KU) in the cofiber sequence and prove these results, we develop a theory of THH and TC of Waldhausen categories and prove the analogues of Waldhausen's theorems for K-theory. We resolve the longstanding confusion about localization sequences in THH and TC, and establish a specialized devissage theorem.

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