A homology theory for ring spectra.
Introduced by Bokstedt in the early 80s.
See Ostvaer for an introduction to maps from algebraic K-theory to Hochschild homology (Dennis trace map), topological Hochschild homology, and topological cyclic homology (cyclotomic trace map). These fit into a commutative diagram with maps from TC to THH and from THH to HH.
arXiv: Experimental full text search
CT (Category theory)?, AT (Algebraic topology)?, AG (Algebraic geometry)?, NCG (Algebra and noncommutative geometry)?
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
Rognes: http://www.math.uiuc.edu/K-theory/118, http://www.math.uiuc.edu/K-theory/119, http://www.math.uiuc.edu/K-theory/120
[arXiv:1006.4347] Topological Hochschild Homology of as a module from arXiv Front: math.AT by Samik Basu Let be an -ring spectrum. Given a map from a space to , one can construct a Thom spectrum, , which generalises the classical notion of Thom spectrum for spherical fibrations in the case , the sphere spectrum. If is a loop space () and is homotopy equivalent to for a map from to , then the Thom spectrum has an -ring structure. The Topological Hochschild Homology of these -ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of
This paper considers the case , , the p-adic -theory spectrum, and . The associated Thom spectrum is equivalent to the mod p -theory spectrum . The map is homotopy equivalent to a loop map, so the Thom spectrum has an -ring structure. I will compute using its description as a Thom spectrum.
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.
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.
nLab page on Topological Hochschild homology