Homotopically trivial actions on aspherical spaces and topological rigidity of free actions

Let ρ : G → Homeo(X) be a homotopically trivial action of a compact commutative Lie group on a connected, finitistic, aspherical topological space. We associate with ρ a certain set of homotopical invariants. Using them we introduce the notion of π1-freeness, π1-conjugacy and π1-effectiveness. We check that ρ is free if and only if it is π1-free. Applying the rigidity theorems of F.T. Farrell and L. Jones we prove that π1-conjugate, homotopically trivial, smooth, and free actions of G on appropriate aspherical manifolds are topologically conjugate. Using this we show that the number of topological conjugacy classes of free and smooth Zk-actions that are homotopic to a given free Zk-action on a closed infrasolvmanifold M it is not greather than krank Z(π1(M))Zk.