Crossed Simplicial Groups and Their Associated Homology

We introduce a notion of crossed simplicial group, which generalizes Connes' notion of the cyclic category. We show that this concept has several equivalent descriptions and give a complete classification of these structures. We also show how many of Connes' results can be generalized and simplified in this framework. A simplicial set (resp. group) is a family of sets (resp. groups) {Gn}n>0 together with maps (resp. group homomorphisms) which satisfy some well-known universal formulas. The geometric realization of a simplicial set is a space and the geometric realization of a simplicial group is a topological group. We define a crossed simplicial group as a simplicial set 6\ = {Gn}n>0 such that the Gn 's are groups and the faces and degeneracies are crossed group homorphisms, that is, satisfy a formula like f(gg') = f(g)(g-f)(g') (see §1 for the precise definition). A simplicial group is thus the particular case of a trivial action. The geometric realization of a crossed simplicial group is still a topological group. The reason for introducing such objects comes from cyclic homology whose definition, as given by Connes in [C], relies on the existence of a certain category A (denoted AC in this paper ) satisfying some special properties. In fact these properties are equivalent to the following assertion: the standard simplicial circle can be endowed with the structure of a crossed simplicial group Ct {Cn}n>0 with Cn = Z/n + l (cyclic groups). In [L] we remarked that the family of dihedral groups {Dn+l}n>0 (resp. quaternion groups {ô„+1}„>o ) forms a crossed simplicial group (but not a simplicial group). The notion of crossed simplicial group provides a useful conceptual framework for studying these basic examples. In this paper we investigate the existence of other families of groups bearing a crossed simplicial group structure. In particular we show that it is the case for the family of symmetric groups St {Sn+l}n>0. Then in 3.6 we give a complete classification theorem: Received by the editors May 28, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 18D05, 18F25, 18G30. © 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page