Singular decompositions of a cap product

Chataur, David; Saralegi-Aranguren, Martintxo; Tanré, Daniel

doi:10.1090/proc/13508

Singular decompositions of a cap product
HTML articles powered by AMS MathViewer

by David Chataur, Martintxo Saralegi-Aranguren and Daniel Tanré PDF

Proc. Amer. Math. Soc. 145 (2017), 3645-3656 Request permission

Abstract:

In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection homology groups. In this work, we show that this classical cap product is compatible with a cap product in intersection (co)homology that we have previously introduced. If the pseudomanifold is also normal, for any commutative ring of coefficients, the existence of a classical Poincaré duality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities.

References

David Chataur, Martintxo Saralegi-Aranguren, and Daniel Tanré, Intersection cohomology. Simplicial blow-up and rational homotopy, arXiv mathematics e-prints 1205.7057v4 (2012). To appear in Mem. Amer. Math. Soc.
—, Dualité de Poincaré et homologie d’intersection, arXiv mathematics e-prints 1603.08773v1 (2016).
—, Homologie d’intersection. Perversités générales et invariance topologique, arXiv mathematics e-prints 1602.03009v1 (2016).
Albrecht Dold, Lectures on algebraic topology, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 200, Springer-Verlag, Berlin-New York, 1980. MR 606196
Greg Friedman, Singular intersection homology. Available at http://faculty.tcu.edu/gfriedman/index.html.
Greg Friedman and James E. McClure, Cup and cap products in intersection (co)homology, Adv. Math. 240 (2013), 383–426. MR 3046315, DOI 10.1016/j.aim.2013.02.017
Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. MR 572580, DOI 10.1016/0040-9383(80)90003-8
Mark Goresky and Paul Siegel, Linking pairings on singular spaces, Comment. Math. Helv. 58 (1983), no. 1, 96–110. MR 699009, DOI 10.1007/BF02564627
Henry C. King, Topological invariance of intersection homology without sheaves, Topology Appl. 20 (1985), no. 2, 149–160. MR 800845, DOI 10.1016/0166-8641(85)90075-6
Clinton Graydon McCrory, POINCARE DUALITY IN SPACES WITH SINGULARITIES, ProQuest LLC, Ann Arbor, MI, 1972. Thesis (Ph.D.)–Brandeis University. MR 2622359
Clint McCrory, Zeeman’s filtration of homology, Trans. Amer. Math. Soc. 250 (1979), 147–166. MR 530047, DOI 10.1090/S0002-9947-1979-0530047-X
L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123–136; ibid. 47 (1972), 137–163. MR 319207, DOI 10.1007/BF02566793
Andrei Verona, Le théorème de de Rham pour les préstratifications abstraites, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A886–A889 (French). MR 290375