Singular decompositions of a cap product

TL;DR

This paper explores the compatibility of the classical cap product with a previously introduced intersection (co)-homology cap product in pseudomanifolds, linking Poincaré duality to intersection homology group isomorphisms.

Contribution

It demonstrates the compatibility of cap products in classical and intersection homology and relates Poincaré duality to intersection homology group isomorphisms in normal pseudomanifolds.

Findings

Classical cap product factorizes through intersection homology in pseudomanifolds.

Compatibility established between classical and intersection (co)-homology cap products.

Poincaré duality is equivalent to intersection homology group isomorphisms for normal pseudomanifolds.

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 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.