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• W2078477162 abstract "Banach manifolds whose tangent bundles admit a reduction to the Fredholm group have been intensively studied in the last few years. Here we show that such a manifold (under appropriate smoothness and separability restrictions) is homotopy equivalent to the union of a nested sequence of closed finite-dimensional submanifolds. Introduction. Banach manifolds whose tangent bundles admit a reduction to the Fredholm group have been intensively studied in the last few years. Here we show that such a manifold (under appropriate smoothness and separability restrictions) is homotopy equivalent to the union of a nested sequence of closed finite-dimensional submanifolds. This result allows one to develop a fairly elaborate algebraic topology of such manifolds and the proper Fredholm maps between such manifolds. In particular, duality theorems, degree theory, and a version of Lefschetz's coincidence theorem have been obtained [9], [10]. In a different direction, this result is the starting point of Eells and Elworthy's theorem that every smooth, paracompact, separable Hilbert manifold is diffeomorphic to an open subset of 12, [4]. 1. Preliminaries. In what follows E, F will always denote infinite-dimensional Banach spaces (over the real numbers) and L(E, F) the space of continuous, linear maps from E to F with the norm topology. We shall denote L(E, E) by L(E). We use the following notation and nomenclature for the subspaces of L(E, F) or L(E), below: C(E, F)= the completely continuous or compact operators in L(E, F). C (E)--=C (E5 E). Lis(E, F)_ the bijective isomorphisms in L(E, F). GL(E)_Lis(E, E), called the general linear group of E. GC(E)-{a e GL(E) I a=I+a'; a' E C(E)}. GC(E) is a closed normal subgroup of GL(E) of index 2 and we shall call it the Fredholm group of E. q)(E, F)_ {a e L(E, F) I dim (Kernel a) < oo, dim (Coker a) < oo}. Received by the editors September 8, 1969. AMS Subject Classifications. Primary 5755." @default.
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• W2078477162 date "1970-02-01" @default.
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• W2078477162 title "The homotopy type of Fredholm manifolds" @default.
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