# Multiplicative de Rham Theorems for Relative and Intersection Space Cohomology

## TL;DR

This paper establishes a ring isomorphism between de Rham and intersection space cohomology for stratified pseudomanifolds, extending Poincaré Duality and providing explicit constructions.

## Contribution

It constructs an explicit de Rham isomorphism that preserves ring structures for intersection space cohomology, a significant advancement over previous graded isomorphisms.

## Key findings

- Proves a ring isomorphism between de Rham and intersection space cohomology.
- Extends Poincaré Duality to stratified pseudomanifolds.
- Provides a new proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds.

## Abstract

We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincar\'e Duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds which we use in the proof of our main result.