Diffeomorphisms of the 2-sphere

The analogue of Theorem A for the topological case was proved by H. Kneser 2. The problem in his case seems to be of a different nature from the differentiable case. J. Munkres 3 has proved that Q is arcwise connected. Conversations with R. Palais have been helpful in the preparation of this paper. Let I2 be the square in the Euclidean plane E2 with coordinate (t, x) such that (t, x) E12 if O r> 1, and that all function spaces considered possess the Cr topology. We further assume that all diffeomorphisms are C. Let I1CI2 denote the subset (t, x) EI21 t= 1, dfp be the differential of a diffeomorphism f at p, and uo be the vector (1, 0) in E2 considered as its own tangent vector space. Then denote by g the space of diffeomorphisms of 12 onto I2 such that if fEG, then (a) f = on some neighborhood of 12I, , and (b) dfp (uo) = uo for all p in some neighborhood of I,.