Dihomology III. A Generalization of the Poincaré Duality for Manifolds

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which is a topological invariant of the space X, but not an invariant of homotopy type. The E term is the Cech cohomology of X with coefficients in the sheaf £ of local singular homology of X, and the ETM term is related to the ordinary global singular homology of X. The sequence therefore relates the local and global structures of the space. If X is a closed orientable w-manifold, then the local homology sheaf reduces to the simple sheaf of integers, and the spectral sequence collapses to the familiar isomorphism of Poincare, H^Hn_Q. If X is a polyhedron, there is a simple way of defining E, and its dual $, using a triangulation. In order to obtain the spectral sequences as quickly as possible we give this simplicial method in §2, and summarize the properties of the sequences in Theorem 1. In §3 we generalize E,£ to arbitrary topological spaces, using a combination of singular homology and Cech cohomology, and verify that the simplicial sequences are in fact topological invariants. The functorial qualities of E,$ are also discussed. Since both E and $ involve both homology and cohomology, they are not functors on the category of topological spaces but we prove in Theorem 2 that if we generalize them further they are functors on a category of maps, the sequences of a space being those associated with the identity map. The last section is concerned with the geometrical interpretation of E and $, and, in particular, of the filtrations induced on the homology and cohomology groups. It appears that the filtration of a homology or cohomology class has something to do with the dimension of that part of the space in which it is 'situated', and we prove two theorems which are feelers in this direction. Theorem 3 connects the homology filtration with cap products, while Theorem 4 concerns the filtration of a cohomology