Homology of linear groups via cycles in BG× X

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Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups Hᵢ(X,G) by considering cycles in the simplicial scheme BG× X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups Hᵢ(Δʲ,G), which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. <p> We also compute some examples, focusing in particular on the case X=Spec(k). In the case where k is the real numbers, there is a connection between the groups Hᵢ and the Z/2-equivariant cohomology of the classifying space of the discrete group G(R).