Cyclic Spaces and S ¹-Equivariant Homology

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 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.