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 fi 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)EI2 ii Ofktfkl and Oflxfkl. Let e: P->P denote the identity diffeomorphism and J, the space of diffeomorphisms, with the Cr topology, of P onto P which agree with e on some neighborhood of P, the boundary of P. The Cr topology is such that two maps are close with respect to it if they are close and their first r derivatives are close. See R. Thom [4] for details. We assume that r is fixed in this paper, 00 ̂ r>l, and that all function spaces considered possess the CT topology. We further assume that all diffeomorphisms are C°°. Let IiEP denote the subset {(t, x)EP\t=l}, dfp be the differential of a diffeomorphism / at p, and u0 be the vector (1, 0) in E2 considered as its own tangent vector space. Then denote by S the space of diffeomorphisms of P onto P such that if /GS, then (a) f=e on some neighborhood of P — Ii, and (b) dfp(u0)=u0 for all p in some neighborhood of h.