Cyclic Spaces and S1-Equivariant Homology
- 388 Downloads
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 1 ≅ Bℤ = |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.
KeywordsConjugation Class Simplicial Group Discrete Group Cyclic Module Geometric Realization
Unable to display preview. Download preview PDF.
Bibliographical Comments on Chapter 7
- Goodwillie, T.G., Cyclic homology, derivations and the free loop space, Topology 2.4 (1985), 187–215.Google Scholar
- 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
- Waldhausen, F., Algebraic K-theory of topological spaces I, Proc. Symp. Pure Math. 32 Ams (1978), 35–60.Google Scholar
- 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
- Blanc, P., Brylinski, J.-L., Cyclic homology and the Selberg principle, J. Funct. Analysis (1992), to appear.Google Scholar