2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61499We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there is a principal fibre bundle isomorphism between pull-backs obtained from homotopic maps. This enables simple proofs of several results on the structure of principal fibre bundles. No new results are obtained---this is simply an accessible presentation of an important idea.5 pages; an introductory note; unpublishedDifferential Geometry55R15text