Homology and homotopy of infinite dimensional manifolds



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This paper is concerned with a study of the structure of infinite dimensional manifolds, giving information about the homology and homotopy, and leading to the construction of a codimensional homology functor which distinguishes sets of finite codimension and which satisfies a Poincare duality with respect to the singular cohomology. Attention is restricted to separable differentiable Hilbert manifolds which are Cauchy and geodesically complete and which support finite dimensional vector valued functions with associated thin singular sets so that these sets can be removed via diffeomorphisms between the manifolds and the complements of the thin subsets. This leads to representations for these manifolds as the inverse limit of finite dimensional manifolds which are the Images of the given manifold under a vector valued function, with the structure of the inverse system being determined by a sequence of foliations and an associated sequence of q-parameter groups of diffeomorphisms. There is also a strong homotopy equivalence between the given infinite dimensional manifold and the direct limit of the above mentioned finite dimensional manifolds, A homology functor H[lowered infinity-p] (•,Z) is then constructed by using the strong homotopy equivalence and the connecting homomorphisms of Mayer-Vletoros exact sequences which arise from splittings of the finite dimensional manifolds used In the representations. The ensuing duality is independent of any representation.