Abstract
This chapter introduces the (integral) topological cyclic homology via a pullback diagram involving invariants of topological Hochschild homology. A lift of the trace to topological cyclic homology is given. Considerations of what occurs when completing at a prime is given special attention, as are connections to more algebraic counterparts like negative cyclic homology.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.F. Adams. On the groups J(X), IV. Topology, 5:21–71, 1966.
C. Ausoni. On the algebraic K-theory of the complex K-theory spectrum. Invent. Math., 180(3):611–668, 2010.
C. Ausoni and J. Rognes. Algebraic K-theory of topological K-theory. Acta Math., 188(1):1–39, 2002.
A.J. Blumberg. A discrete model of S 1-homotopy theory. J. Pure Appl. Algebra, 210(1):29–41, 2007.
M. Bökstedt, W.C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3):465–539, 1993.
M. Bökstedt and I. Madsen. Topological cyclic homology of the integers. Astérisque, 226(7–8):57–143, 1994. K-theory (Strasbourg, 1992).
M. Bökstedt and I. Madsen. Algebraic K-theory of local number fields: the unramified case. In Prospects in Topology, Princeton, NJ, 1994, volume 138 of Ann. of Math. Stud., pages 28–57. Princeton University Press, Princeton, NJ, 1995.
D. Burghelea, Z. Fiedorowicz, and W. Gajda. Power maps and epicyclic spaces. J. Pure Appl. Algebra, 96(1):1–14, 1994.
G. Carlsson. Equivariant stable homotopy and Segal’s Burnside ring conjecture. Ann. Math. (2), 120(2):189–224, 1984.
A. Connes. Cohomologie cyclique et foncteurs Extn. C. R. Acad. Sci. Paris Sér. I Math., 296(23):953–958, 1983.
A. Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985.
W.G. Dwyer, M.J. Hopkins, and D.M. Kan. The homotopy theory of cyclic sets. Trans. Am. Math. Soc., 291(1):281–289, 1985.
T.G. Goodwillie. Cyclic homology, derivations, and the free loopspace. Topology, 24(2):187–215, 1985.
T.G. Goodwillie. Relative algebraic K-theory and cyclic homology. Ann. Math. (2), 124(2):347–402, 1986.
T.G. Goodwillie. Letter to F. Waldhausen, August 10 1987.
T.G. Goodwillie. Notes on the cyclotomic trace. Lecture notes for a series of seminar talks at MSRI, Spring 1990, December 1991.
J.P.C. Greenlees and J.P. May. Generalized Tate cohomology. Mem. Am. Math. Soc., 113(543):178, 1995.
L. Hesselholt. On the p-typical curves in Quillen’s K-theory. Acta Math., 177(1):1–53, 1996.
L. Hesselholt. Witt vectors of non-commutative rings and topological cyclic homology. Acta Math., 178(1):109–141, 1997.
L. Hesselholt. Correction to: “Witt vectors of non-commutative rings and topological cyclic homology” [Acta Math. 178 (1997), no. 1, 109–141]. Acta Math., 195:55–60, 2005.
L. Hesselholt and I. Madsen. Cyclic polytopes and the K-theory of truncated polynomial algebras. Invent. Math., 130(1):73–97, 1997.
L. Hesselholt and I. Madsen. On the K-theory of finite algebras over Witt vectors of perfect fields. Topology, 36(1):29–101, 1997.
L. Hesselholt and I. Madsen. On the K-theory of local fields. Ann. Math. (2), 158(1):1–113, 2003.
J.D.S. Jones. Cyclic homology and equivariant homology. Invent. Math., 87(2):403–423, 1987.
L.G. Lewis Jr., J.P. May, M. Steinberger, and J.E. McClure. Equivariant Stable Homotopy Theory, volume 1213 of Lecture Notes in Mathematics. Springer, Berlin, 1986. With contributions by J.E. McClure.
J.-L. Loday. Cyclic Homology, 2nd edition, volume 301 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1998. Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili.
J.-L. Loday and D. Quillen. Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helv., 59(4):569–591, 1984.
I. Madsen. Algebraic K-theory and traces. In Current Developments in Mathematics, Cambridge, MA, 1995, pages 191–321. International Press, Cambridge, 1994.
M.A. Mandell and J.P. May. Equivariant orthogonal spectra and S-modules. Mem. Am. Math. Soc., 159(755):108, 2002.
J. Rognes. Algebraic K-theory of the two-adic integers. J. Pure Appl. Algebra, 134(3):287–326, 1999.
J. Rognes. The smooth Whitehead spectrum of a point at odd regular primes. Geom. Topol., 7:155–184, 2003.
S. Schwede and B. Shipley. Equivalences of monoidal model categories. Algebr. Geom. Topol., 3:287–334, 2003.
G.B. Segal. Equivariant stable homotopy theory. In Actes du Congrès International des Mathématiciens, Tome 2, Nice, 1970, pages 59–63. Gauthier-Villars, Paris, 1971.
J.-P. Serre. Corps locaux. In Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, volume 1296 of Actualités Scientifiques Et Industrielles. Hermann, Paris, 1962.
J. Spaliński. Strong homotopy theory of cyclic sets. J. Pure Appl. Algebra, 99(1):35–52, 1995.
A.A. Suslin and M. Wodzicki. Excision in algebraic K-theory. Ann. Math. (2), 136(1):51–122, 1992.
T.T. Dieck. Orbittypen und äquivariante Homologie, I. Arch. Math. (Basel), 23:307–317, 1972.
T.T. Dieck. Orbittypen und äquivariante Homologie, II. Arch. Math. (Basel), 26(6):650–662, 1975.
S. Tsalidis. On the topological cyclic homology of the integers. Am. J. Math., 119(1):103–125, 1997.
S. Tsalidis. Topological Hochschild homology and the homotopy descent problem. Topology, 37(4):913–934, 1998.
B.L. Tsygan. Homology of matrix Lie algebras over rings and the Hochschild homology. Usp. Mat. Nauk, 38(2(230)):217–218, 1983.
M. Weiss and B. Williams. Automorphisms of manifolds and algebraic K-theory, II. J. Pure Appl. Algebra, 62(1):47–107, 1989.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
Dundas, B.I., Goodwillie, T.G., McCarthy, R. (2013). Topological Cyclic Homology. In: The Local Structure of Algebraic K-Theory. Algebra and Applications, vol 18. Springer, London. https://doi.org/10.1007/978-1-4471-4393-2_6
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4393-2_6
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4392-5
Online ISBN: 978-1-4471-4393-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)