C 1 Actions of Compact Lie Groups on Compact Manifolds are C 1 -Equivalent to C ∞ Actions

Abstract

Introduction. Let C. and C2 be two categories and let F: C,---O#C2 be a weakening of structure ( forgetful ) functor, i. e. for each pair of objects X1 and X2 of C, the map of Morph (X1, 2X) to Morph(F(X1), F(X2)) is injective. Let 01 and 02 denote the equivalence classes of objects of C, and C2 defined by the equivalence relations of isomorphisms in the respective categories. Then F induces a map F: 01O2 and the question of whether X is injective, surjective, or bijective is often of interest. For example for O ? kc ?oo let Mank denote the category of Ck paracompact manifolds and Ck maps, and for k > I let FZk: Mank Manz be the obvious forgetful functor. A Then for I = 0 Fok: Mank -Man0 is neither surjective nor injective, i.e. there exist topological manifolds which admit no Uk structure [3] and there exist topological manifolds with non-isomorphic Ck structures [5]. However if I> 0 then Flk is always bijective, i.e. every CL manifold admits a compatible Ck structure [10] and if two Ck manifolds are CL diffeomorphic they are Ck diffeomorphic (the latter is trivial from standard approximation theorems).