The homotopy type of the space of diffeomorphisms. I

A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result Diff ( D n , ∂ ) ≃ Ω n + 1 ( Top n / O n ) {Diff}({D^n},∂ ) {Ω n + 1}({Top_n}/{O_n}) is obtained. The main technique is fibrewise smoothing.