Diffeomorphisms for Hilbert Manifolds and Handle Decomposition

1. We announce here the following result: Two homotopic diffeomorphisms of a paracompact separable Hubert manifold of infinite dimension are isotopic. (1) In this paper, a hilbert manifold (A-manifold) with or without boundary is always hausdorff, paracompact, separable C°°-differentiable and with the infinite dimensional separable hilbert space H as local model. Let M(M, dM) be an /^-manifold (with boundary), X(X, dX), an A-manifold or finite dimensional manifold (with boundary). (a) A closed imbedding 0:X~>ikT(: (X, 3X)->(Af, dM)) is a C°°-injective map :X—>M, such that the differential d*(x) is injective for any x, and (M) is closed (for the case with boundary we ask more, (j>~~{dM) =dM and is transversal to dM in dM). (b) A closed tubular neighborhood of a closed imbedding of infinite codimension, :X-+M, (:(X, dX)—>(M, dM)), is a closed imbedding$:XXD»-*M($:(X, dX)XD«>->(M, dM)) which extends to an open imbedding ^ : I X H > M ( ^ ( I , dX)XH->(M, dM) with $-*(dM)=dXXH). REMARKS. (1) Any closed imbedding of infinite codimension has closed tubular neighborhoods [3]. (2) For <£i and $2 two closed tubular neighborhoods of a closed imbedding :X—>ikf($:(X, dX)-±(M, dM)), there exists an isotopy ht:M-+M, (ht:(M, dM)->(M, dM)), O g / g l , such that h0 = id, ht'$=([> and hi*$i = $2 [2, Theorem 4.1]. By an isotopy as in [2], we mean a level preserving C^-diffeomorphism h:MXI—*MXI, (h:(M, dM)XI->(M, dM)Xl), i.e., h(x, 0 = (**(*). /). (c) Let M(M, dM) be an A-manifold (with boundary); A closed imbedded submanifold with boundary {A, dA), such that ^4CInt M and A\dA is open submanifold of M, is called a zero-codimensional closed submanifold (O-c-submanifold). The O-c-submanifold (B, dB) is called a collar neighborhood of the O-c-submanifold (A, dA), if AC Int B and ( 5 \ I n t A, d(B\lnt A)) is diffeomorphic to (oil X [0, l],dAXd[0, l ] ) . 1 The author was partially supported by the National Science Foundation.