Characteristic classes for modules over groups. I

Abstract

Suppose <£■ is a group and M is a module over the integral group ring of $.Then the homology groups, H^M), are also O-modules.The usual method of seeing this is to use the standard resolution, S", of Z for M because the summands, S¡, are O-modules themselves, and the boundary map is a <D-homomorphism.However, this complex is a very cumbersome one, and one would like to see if the action of <t on H*(M) can be obtained from an arbitrary resolution.Let £>* be any resolution of Z for M. The obvious question to ask is whether one can make D, into a <P-module in any natural way.One of the results of this paper is to show that this can not be done in general.It is the obstructions to this which give rise to the characteristic classes of the title.The first section develops the notion of a ^-system for £>* which is an approximation to an action of <P on D¡.One part of a O-system is, for each / and a e <P, a Z-homomorphism A^o): D¡ -*■ D¡ which is a chain map and satisfies an appropriate semilinearity condition.The point is that At(a) ° Aí(t)^A¡(ot) in general.It is easily seen, however, that they are chain homotopic, and it is such a chain homotopy Ufa, t) between Afa) ° A¡(t) and AXot) which is the other part of a <P-system and measures the obstruction to the existence of an action of O on Z)¡.If the complex D* is reasonable, these obstructions can be described as follows: For each a, r e <P, í/¡(<7, r) e Horn (£>¡, Di+X) defines an element in Hom(/7f(M),//i+i(M))which is H'(M, Hi + X(M)) if the final (unwritten) coefficients are nice enough.This can be thought of as defining a nonhomogeneous 2-cochain aii + 1 for <P with coefficients in H\M, Hi + X(M)), coi + 1 turns out to be a cocycle and its cohomology class vi + 1(M) e /F2((P, H\M, Hi+X(M))) is what we call the /th characteristic class of M. v'(M) depends only on <P, M, and the action of <D on M.These algebraic characteristic classes satisfy a naturality condition similar to the one satisfied by topological (e.g.Stiefel-Whitney) ones, and if M is Z-free, there is an analogue to the Whitney sum theorem.