nLab
topological cyclic homology

Topological Hochschild (resp. topological cyclic) homology is an adaptation of Hochschild (resp. cyclic) homology to the setup of ring spectra. They are introduced in

• M. Bökstedt, Topological Hochschild homology, Bielefeld, 1985, 1988
• M. Bökstedt, W.C. Hsiang, Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 463-539, MR94g:55011, doi
• M. Bökstedt, I. 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

• Bjørn Ian Dundas, Randy McCarthy, Topological Hochschild homology of ring functors and exact categories, J. Pure Appl. Algebra 109 (1996), no. 3, 231–294, MR97i:19001, doi

These theories are the target for the trace map from $K$-theory, so they can be viewed as an approximation to algebraic K-theory of a ring spectrum.

• 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 $Z/p^{k}Z$, 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, pdf
• 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
• Thomas Geisser, Motivic Cohomology, K-Theory and Topological Cyclic Homology, Handbook of K-theory II.1, pdf
• R. McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222.
• Ricardo Andrade, THH notes, MIT juvitop seminar pdf, babytop seminar pdf
• Anatoly Preygel, Hochschild homology notes, juvitop seminar, pdf
• Gunnar Carlsson, Christopher L. Douglas, Bjørn Ian Dundas, Higher topological cyclic homology and the Segal conjecture for tori, Adv. Math. 226 (2011), no. 2, 1823–1874, MR2737802, doi
• Ib Madsen, Algebraic K-theory and traces, pdf
• Dundas, Goodwillie and McCarthy, The local structure of algebraic K-theory, book in progress, pdf
• J. McClure, R. Staffeldt, On the topological Hochschild homology of $bu$, I, pdf
• Daniel Joseph Vera, Topological Hochschild homology of twisted group algebra, MIT Ph. D. thesis 2006, pdf