Topological Hochschild homology

General Introduction
A homology theory for ring spectra.
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Search results
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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 Ktheory handbook.
Bokstedt: Topological Hochshild homology. Preprint, 1985.
Waldhausen et al in JLMS: THH article
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
[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 padic $K$theory spectrum, and $\zeta = 1p \in \pi1BGL1Kp^\wedge$. 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 AusoniRognes, 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(kuKU) 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 Ktheory. We resolve the longstanding confusion about localization sequences in THH and TC, and establish a specialized devissage theorem.
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