# A commutative model for PL compactly supported cohomology in characteristic zero

## TL;DR

This paper extends classical rational homotopy theory to a compactly supported setting in characteristic zero, providing a simplicial model for singular cohomology with potential for new research directions.

## Contribution

It introduces a simplicial version of the compactly supported de Rham complex that models singular compactly supported cohomology in characteristic zero.

## Key findings

- Provides a simplicial model for compactly supported cohomology
- Extends Sullivan's rational homotopy theory to compact support
- Opens new avenues in proper homotopy theory

## Abstract

We show that that classical rational homotopy theory in the sense of Sullivan [6] can be extended compactly supported setting. This presents a simplicial version of the compactly supported de Rham complex in characteristic zero, and proving that it models singular compactly supported cohomology. This presents new avenues of possible study in proper homotopy theory, and further extensions of the ideas within the classical literature of Quillen [5] and Sullivan.