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

Revised on October 6, 2011 00:32:32
by Zoran Škoda
(161.53.130.104)