On Bredon's trick: from local to global properties

TL;DR

This paper reviews Bredon's trick, a lemma that extends local properties to global ones on topological spaces, and demonstrates its applications in De Rham cohomology and stratified pseudo-manifolds.

Contribution

It provides a comprehensive review of Bredon's trick and illustrates its utility in deriving classical results and analyzing complex topological structures.

Findings

Bredon's trick effectively extends local properties to global contexts.

Alternative proofs of classical results in De Rham cohomology are achieved.

The trick is fundamental in studying stratified pseudo-manifolds.

Abstract

One of the most interesting problems that arise when studying certain structures on topological spaces and in particular on differential manifolds, is to be able to extend the properties that are valid locally to the whole space. A useful tool, which has perhaps been underestimated, is a lemma introduced by G. Bredon, which we refer to as Bredon's trick and which allows the extension of local properties to certain topological spaces. We make a review of this result and show its application in the context of De Rham's cohomology, we will see how this trick allows to give natural alternative demonstrations to classic results, as well as it is fundamental in other cases such as in stratified pseudo-manifolds.