Automorphisms of manifolds and algebraic K-theory: I

We investigate the homotopy type of $$\widetilde{TOP}$$ (M)/TOP(M), where M is a compact manifold, TOP(M) is the simplicial group of homeomorphisms of M which restrict to the identity on ∂M, and $$\widetilde{TOP}(M)$$ is the simplicial group of block homeomorphisms of M which restrict to the identity on ∂M. In the so-called topological concordance stable range of M, we obtain an expression in terms of the topological Whitehead spectrum of M. If M is smooth, we also investigate the homotopy type of $$\widetilde{DIF}F$$ (M)/DIFF(M); in the smooth concordance stable range of M, it has an expression in terms of the smooth Whitehead spectrum of M.