Cech homology and the Novikov Conjectures for K- and L-Theory

In [5], we studied the assembly map in algebraic Kand L-theory, and showed that the assembly map splits for a class of groups Γ with finite BΓ for which EΓ admits a metrizable, contractible, equivariant compactification such that the Γ-action is “small at infinity”. This means that every compact subset of EΓ when translated out near a point in the boundary becomes small i. e. for every y ∈ ∂EΓ and for every neighborhood U of y in EΓ, there is a neighborhood V of y so that γK ∩ V 6= ∅ implies γK ⊂ U . The method used in [5] was to use continuously controlled Kand L-theory. Given a spectrum S one may define homology with coefficients in the spectrum S by the formula h∗(X, x0;S) = π∗(X ∧ S)