nLab intersection cohomology

Idea

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.

References

• Markus Banagl, Topological invariants of stratified spaces, Springer Monographs in Mathematics, Springer (2000) [doi:10.1007/3-540-38587-8]

  (on stratified spaces)

• Markus Banagl, Rational generalized intersection homology theories, Homology, Homotopy and Applications, 12 1 (2010) 157-185 [pdf, euclid]

• Robert MacPherson, Intersection homology and perverse sheaves, 1990 notes pdf

• R. MacPherson, Global questions in the topology of singular spaces, Proc. ICM Warsaw 1983, pdf

• M. Goresky, G. MacPherson, Morse theory and intersection homology theory, Astérisque 101-102 parts II-III (1983), 135-192.

• 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

• Steven L. Kleiman, The development of intersection homology theory, Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson. Part 3, 225–282, MR2008e:55006

• T. A. Springer, Quelques applications de la cohomologie d’intersection, Séminaire Bourbaki 24 (1981-1982), Exp. No. 589, 25 p. MR689533, Zbl0526.22014

• George Lusztig, Intersection cohomology methods in representation theory, Proc. ICM 1990, pdf

In the generality of matroids:

• June Huh, Combinatorics and Hodge theory, Proc. Int. Cong. Math. 1 (2022)

  [doi:10.4171/ICM2022/205, pdf, pdf]