While for manifolds there is a Poincaré duality relating homology and cohomology, it fails for singular spaces. One can extend the idea of duality by taking into account (co)dimensions of singular strata. This idea can be concretized in a definition of intersection cohomology which has an additional parameter called perversity. In two special cases of perversity, it boils down to either homology or cohomology; thus some call it intersection homology. Main applications are in study of topology of singular spaces and in geometric representation theory. In particular properties of intersection of cycles of various codimension, including transversality are neatly accounted for.
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wikipedia: intersection homology
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MO question: what happened to intersection homology
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