# Intersection homology duality and pairings: singular, PL, and sheaf-theoretic

## TL;DR

This paper establishes the equivalence of various intersection homology duality theories and product structures across sheaf-theoretic, singular, and PL frameworks, unifying different approaches in intersection homology.

## Contribution

It demonstrates natural isomorphisms between sheaf-theoretic and singular intersection homology dualities and products, including PL and de Rham versions, unifying multiple perspectives.

## Key findings

- Sheaf-theoretic and singular Poincare duality are isomorphic.
- Canonical isomorphisms between intersection cohomology cup product and sheaf pairing.
- De Rham isomorphism preserves product structures.

## Abstract

We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky-MacPherson PL intersection product. We also show that the de Rham isomorphism of Brasselet, Hector, and Saralegi preserves product structures.