Stratified singular spaces and generalized geometric Poincaré complexes

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Poincaré duality is a cornerstone of classical manifold theory. The ordinary cohomology of spaces with singularities does not enjoy this duality. One solution is intersection cohomology, which has a series of drawbacks: much of the internal algebraic structure available in the ordinary cohomology of a space is lacking. It is not stable under deformation of singularities. Its chain-theoretic definition makes it hard to define generalized intersection cohomology theories. Furthermore, it is the wrong theory to describe massless fields in type IIB string theory. In previous work we have associated to certain singular spaces their intersection space, in such a way that the ordinary rational cohomology of the intersection space satisfies Poincaré duality across complementary perversities. In particular, one obtains a new cohomology theory for stratified spaces. We propose to investigate the internal algebraic structure of this new theory, its natural domain of definition, its stability under smooth deformation of singularities, its relevance vis-à-vis type IIB string theory, and its associated spectrum cohomology, e.g. K-theory.