While for smooth manifolds there is a Poincaré duality relating singular homology and singular cohomology, this 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.
Markus Banagl, Topological invariants of stratified spaces, Springer Monographs in Mathematics, Springer (2000) [doi:10.1007/3-540-38587-8]
(on stratified spaces)
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Greg Friedman, Singular Intersection Homology, Cambridge University Press 2020 (doi:10.1017/9781316584446, pdf)
Wikipedia, Intersection homology
F. Kirwan, Jonathan Woolf, An introduction to intersection homology, Chapman & Hall/CRC, 2006. MR2006k:32061
A. Borel et al. Intersection cohomology, Notes on the seminar held at the University of Bern, 1983. Progress in Math. 50, Birkhäuser 1984 MR88d:32024
Springer Online Enc. of Math., intersection homology
MO question: what happened to intersection homology
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George Lusztig, Intersection cohomology methods in representation theory, Proc. ICM 1990, pdf
In the generality of matroids:
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