symmetric monoidal (∞,1)-category of spectra
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Topological Hochschild homology (resp. topological cyclic homology) (see the survey in (May)) is a refinement of Hochschild homology/cyclic homology from commutative rings/algebras to the higher algebra of ring spectra/E-∞ rings/E-∞ algebras.
One motivation for their study comes from computational considerations, as in certain cases, these invariants are easier to compute than algebraic K-theory, while there is a natural morphism of spectra
from the algebraic K-theory spectrum to the topological Hochschild homology spectrum, called the Dennis trace map, whose fiber is relatively well-understood. Since Hochschild homology spectra are naturally cyclotomic spectra, this map factors through the topological cyclic homology spectrum via a map called the cyclotomic trace, which acts much like a Chern character map for algebraic K-theory.
The spectra $THH(R)$ and $TC(R)$ are typically easier to analyze than $K(R)$. Moreover, the difference between them and $K(R)$ is “locally constant” (Dundas-Goodwillie-McCarthy13) and often otherwise bounded in complexity. Accordingly, $THH$ and $TC$ are in practice computationally useful approximations to $K$.
There are various generalizations:
Just as for basic Hochschild homology, there is higher topological Hochschild homology (Carlsson-Douglas-Dundas 08) given not just by derived loop spaces but by derived mapping spaces out of higher dimensional tori.
Just as algebraic K-theory generalizes from E-∞ rings to stable ∞-categories, so do $TC$ and the cyclotomic trace map (Blumberg-Gepner-Tabuada 11)
The original references are
Marcel Bökstedt, Topological Hochschild homology, Bielefeld, 1985, 1988
Marcel Bökstedt, W.C. Hsiang, Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 463-539, MR94g:55011, doi
Marcel Bökstedt, Ib Madsen, Topological cyclic homology of the integers, $K$-theory (Strasbourg, 1992). Astérisque 226 (1994), 7–8, 57–143.
and a further generalization is defined in
Higher topological Hochschild Homology is discussed in
A general abstract construction is in
Review and exposition includes
Peter May, Topological Hochschild and Cyclic Homology and Algebraic K-theory (pdf)
Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, chapter IX of Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)
Bjørn Dundas, Thomas Goodwillie, Randy McCarthy, The local structure of algebraic K-theory, Springer 2013 (pdf)
Teena Gerhardt, Computations in algebraic K-theory, talk at CUNY Workshop on differential cohomologies 2014 (video recording)
Ricardo Andrade, THH notes, MIT juvitop seminar pdf, babytop seminar pdf
Anatoly Preygel, Hochschild homology notes, juvitop seminar, pdf
Thomas Geisser, Motivic Cohomology, K-Theory and Topological Cyclic Homology, Handbook of K-theory II.1, pdf
Ib Madsen, Algebraic K-theory and traces, pdf
Abstract characterization of the Dennis trace and cyclotomic trace is discussed in
An approach using only homotopy-invariant notions, which gives a construction of topological cyclic homology based on a new definition of the ∞-category of cyclotomic spectra is in
See also
T. Pirashvili, F. Waldhausen, Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra 82 (1992), 81-98, MR96d:19005, doi
T. Pirashvili, On the topological Hochschild homology of $\mathbf{Z}/p^k\mathbf{Z}$, Comm. Algebra 23 (1995), no. 4, 1545–1549, MR97h:19007, doi
Z. Fiedorowicz, T. Pirashvili, R. Schwänzl, R. Vogt, F. Waldhausen, Mac Lane homology and topological Hochschild homology, Math. Ann. 303 (1995), no. 1, 149–164, MR97h:19007, doi
Bjørn Ian Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), 223-242, (publisher)
Thomas Geisser, Lars Hesselhoft, Topological cyclic homology of schemes, in: Algebraic $K$-theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math. 67, Amer. Math. Soc. 1999, MR2001g:19003; K-theory archive
R. McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222.
J. McClure, R. Staffeldt, On the topological Hochschild homology of $b u$, I, pdf
Daniel Joseph Vera, Topological Hochschild homology of twisted group algebra, MIT Ph. D. thesis 2006, pdf
V. Angeltveit, A. Blumberg, T. Gerhardt, M. Hill, T. Lawson, M. Mandell, Topological cyclic homology via the norm (arXiv:1401.5001)
THH and TC specifically of ku and ko is discussed in
Christian Ausoni, John Rognes, Algebraic K-theory of topological K-theory, Acta Mathematica March 2002, Volume 188, Issue 1, pp 1-39 (KTheory 0405)
Vigleik Angeltveit, Michael Hill, Tyler Lawson, Topological Hochschild homology of $\ell$ and $ko$ (arXiv:0710.4368)
Andrew Blumberg, Michael Mandell, Localization for $THH(ku)$ and the topological Hochschild and cyclic homology of Waldhausen categories (arXiv:1111.4003)
and of tmf in