Geometric Poincar\'e Lemma

TL;DR

This paper presents a geometric version of the Poincaré Lemma applicable to differential chains, leading to new insights and generalizations of classical theorems like the Intermediate Value and Rolle's Theorems.

Contribution

It introduces a geometric Poincaré Lemma for differential chains, extending classical results to a broader topological and geometric context.

Findings

Every differential cycle with compact support in a contractible open set is a boundary.

The results generalize classical theorems such as the Intermediate Value and Rolle's Theorems.

The lemma applies to the topological vector space of differential chains.

Abstract

A geometric version of the Poincar\'e Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is the boundary of a differential (k+1) -chain with compact support in U. Applications include generalizations of the Intermediate Value Theorem and Rolle's Theorem.