Obstructions to extending diffeomorphisms

Abstract

PROOF. We assume, without loss of generality, that the restriction of F to some neighborhood of Mo is a diffeomorphism. (One may justify this as follows: Consider the manifold obtained from M and MoXI by identifying xEMo with (x, 0) Mo XI; it is diffeomorphic to M. Obtain a manifold from N and No X I similarly. The map which equals F on M and equals the trivial extension of f on Mo XI is a combinatorial equivalence between these manifolds, and its restriction to Mo X (O, 1] is a diffeomorphism.) Restrict F to the manifold M' = M- Mo; it will be a combinatorial equivalence between this manifold and N'=N-No. We apply our obstruction theory (in particular, 5.7 of [2]) to the problem of smoothing this map; our object is to obtain a diffeomorphism of M' onto N' which equals F in a neighborhood of Mo. Such a diffeomorphism will give the required extension of f. Now F: M'->N' is a diffeomorphism mod the (n - 1)-skeleton of M'; we denote it by Fn_-v As an induction hypothesis, assume Fm is a