Diffeomorphism groups of non-compact manifolds endowed with the Whitney C∞C∞-topology

Suppose M is a non-compact connected n  -manifold without boundary, D(M)D(M) is the group of C∞C∞-diffeomorphisms of M   endowed with the Whitney C∞C∞-topology and D0(M)D0(M) is the identity connected component of D(M)D(M), which is an open subgroup in the group Dc(M)⊂D(M)Dc(M)⊂D(M) of compactly supported diffeomorphisms of M  . It is shown that D0(M)D0(M) is homeomorphic to N×R∞N×R∞ for an l2l2-manifold N   whose topological type is uniquely determined by the homotopy type of D0(M)D0(M). For instance, D0(M)D0(M) is homeomorphic to l2×R∞l2×R∞ if n=1,2n=1,2 or n=3n=3 and M is orientable and irreducible. We also show that for any compact connected n-manifold N with non-empty boundary ∂N   the group D0(N∖∂N)D0(N∖∂N) is homeomorphic to D0(N;∂N)×R∞D0(N;∂N)×R∞, where D0(N;∂N)D0(N;∂N) is the identity component of the group D(N;∂N)D(N;∂N) of diffeomorphisms of N that do not move points of the boundary ∂N.