Cyclic Homology pp 223-252

# Cyclic Spaces and S1-Equivariant Homology

Chapter
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 301)

## Abstract

There are several ways of constructing simplicial models for the circle S 1. The simplest one consists in taking only two non-degenerate cells: one in dimension 0 and one in dimension 1. Another model consists in taking the nerve of the infinite cyclic group ℤ. Then its geometric realization has many cells. A priori this latter version, though more complicated in terms of cell decomposition, has the advantage of taking care of the group structure of S 1Bℤ = |B.ℤ| because the nerve B.ℤ, of ℤ is a simplicial group. The main point about the cyclic setting is that in the 2-non-degenerate cell decomposition of S 1 there is a way of keeping track of its group structure. Indeed the corresponding simplicial set has n + 1 simplices in dimension n and there is a canonical identification with the elements of the cyclic group ℤ/(n + 1)ℤ. Then one can recover the group structure on the geometric realization S 1 from the group structure of the cyclic groups.

## Keywords

Conjugation Class Simplicial Group Discrete Group Cyclic Module Geometric Realization
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Bibliographical Comments on Chapter 7

1. Burghelea, D., Fiedorowicz, Z., Cyclic homology and algebraic K-theory of spaces II, Topology 25 (1986), 303–317
2. Goodwillie, T.G., Cyclic homology, derivations and the free loop space, Topology 2.4 (1985), 187–215.Google Scholar
3. Hood, C., Jones, J.D.S., Some algebraic properties of cyclic homology groups, K-theory 1 (1987), 361–384
4. Carlsson, G.E., Cohen, R.L., Goodwillie, T., Hsiang, W.-C., The free loop space and the algebraic K-theory of spaces, K-theory 1 (1987), 53–82
5. ViguÉ-Poirrier, M., Burghelea, D., A model for cyclic homology and algebraic K-theory of 1-connected topological spaces, J. Diff. Geom. 22 (1985), 243–253.Google Scholar
6. Waldhausen, F., Algebraic K-theory of topological spaces I, Proc. Symp. Pure Math. 32 Ams (1978), 35–60.Google Scholar
7. Waldhausen, F., Algebraic K-theory of topological spaces II, Springer Lect. Notes in Math. 1051 (1984), 173–196.
8. Fiedorowicz, Z., Loday, J.-L., Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), 57–87
9. Burghelea, D., The cyclic homology of the group rings, Comment. Math. Helv. 60 (1985), 354–365.
10. Karoubi, M., Homologie cyclique et régulateurs en K-théorie algébrique, C. R. Acad. Sci. Paris Sér. A-B 297 (1983), 557–560
11. Manin, Y.I., Topics in noncommutative geometry, Princeton Univ. Press, 1991. Marciniak, Z., Cyclic homology and idempotent in group rings. Springer Lect. Notes in Math. 1217 (1985), 253–257.Google Scholar
12. Blanc, P., Brylinski, J.-L., Cyclic homology and the Selberg principle, J. Funct. Analysis (1992), to appear.Google Scholar