Symmetric Homology of Algebras

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two spectral sequences for computing symmetric homology are constructed. The relation to cyclic homology is discussed and a couple of conjectures towards further work are proposed. 2000 MSC: 16E40, 55P45, 55S12 Symmetric homology is the analog of cyclic homology, where the cyclic groups are replaced by symmetric groups. The second author and Loday [5] developed the notion of crossed simplicial group as a framework for making this idea precise. Definition 1 A crossed simplicial group is a category ∆G whose objects are the sets [n] = {0, 1, 2, . . . , n} for n ≥ 0, which contains the simplicial category ∆, and such that any morphism [m] → [n] factors uniquely as [m] ∼= −→ [m] γ −→ [n], where γ is a morphism in ∆. The collection of groups {Gn = Aut∆G([n]) }n≥0 are called the underlying groups of ∆G. The commutation relations implicit in ∆G endow {Gn}n≥0 with the structure of a simplicial set (but not necessarily the structure of a simplicial group).