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An Introduction to Pull-backs of Bundles and Homotopy Invariance

Scott Morrison
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Abstract

We 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.

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Scott Morrison. 2001-05-19. An Introduction to Pull-backs of Bundles and Homotopy Invariance. https://arxiv.org/abs/math/0105161

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