Singular decompositions of a cap-product

David Chataur (1) , Martintxo Saralegi-Aranguren (2) , Daniel Tanré (3)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
2 LML - Laboratoire de Mathématiques de Lens
3 LPP - Laboratoire Paul Painlevé

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. As a corollary, for any commutative ring of coefficients, the existence of a classical Poincar´e duality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities. Our results answer a question asked by G. Friedman.

Domains: Mathematics [math]; Algebraic Topology [math.AT]

David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré. Singular decompositions of a cap-product. Proceedings of the American Mathematical Society, 2017, 145 (8), pp.3645-3656. ⟨hal-01332473⟩