Poincar\'e duality with cap products in intersection homology

TL;DR

This paper establishes a Poincaré duality via cap products in intersection homology for any commutative ring, using a geometric blow-up process, extending previous results that required additional hypotheses.

Contribution

It introduces a blown-up cochain complex for intersection cohomology that works over any commutative ring and proves its topological invariance under certain conditions.

Findings

Poincaré duality via cap product is achieved without torsion restrictions.

The blown-up intersection cohomology is topologically invariant for certain pseudomanifolds.

The framework accommodates general perversities and large perversities.

Abstract

For having a Poincar\'e duality via a cap product between the intersection homology of a paracompact oriented pseudomanifold and the cohomology given by the dual complex, G. Friedman and J. E. McClure need a coefficient field or an additional hypothesis on the torsion. In this work, by using the classical geometric process of blowing-up, adapted to a simplicial setting, we build a cochain complex which gives a Poincar\'e duality via a cap product with intersection homology, for any commutative ring of coefficients. We prove also the topological invariance of the blown-up intersection cohomology with compact supports in the case of a paracompact pseudomanifold with no codimension one strata. This work is written with general perversities, defined on each stratum and not only in function of the codimension of strata. It contains also a tame intersection homology, suitable for large perversities.