Smoothings and Homeomorphisms for Hubert Manifolds by Dan Burghelea and David Henderson
نویسندگان
چکیده
0. The aim of this paper is to announce several results concerning smoothing Hubert manifolds and some of their consequences for homeomorphisms. All manifolds are assumed to be Hausdorff, paracompact, and separable with local model the oo-dimensional separable Hubert space H. (All separable infinite dimensional Fréchet spaces are homeomorphic to H.) As usual, we denote by DTM the unit disc and by S°° the unit sphere in H. A pair (M, dM) is called a differentiable manifold with boundary if it has a C°°-differentiable structure with local model the Hubert half space HX [0, oo). The boundary dM, is the set of those points which are mapped onto HX{0} by some chart. It can be shown that dM is a differentiable manifold and is locally collared and thus collared in M. (See for instance, the existence of closed tubular neighborhoods [ll].) This definition is not adequate for topological manifolds with boundary, because by Klee [10], JEZ"X[0, oo) is homeomorphic to H. We thus define a topological manifold with boundary to be a pair (M, dM) such that (a) M and dM are topological manifolds, (b) dM is collared in M, or equivalently locally collared (according to M. Brown, Topology of Z-manifolds and related topics, edited by M. K. Fort, p. 88). Our results are the following:
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