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FALL 2012 MATH 8230 (VECTOR BUNDLES) LECTURE …

FALL 2012 math 8230 ( vector BUNDLES) LECTURE NOTES. 1. DEFINITIONS: vector BUNDLES AND STRUCTURE GROUPS. A vector bundle over a topological space M (or with base space M ) is, essentially, family of vector spaces continuously parametrized by M . (I'm using the letter M to denote the base space of the vector bundle as a concession to the fact that in most of the applications we'll be interested in the base space will be a smooth manifold; however for basic definitions and results it there is no need to restrict to this case.) One way of making this precise is as follows: Definition A (real, rank-k) vector bundle over a topological space M is a continuous map : E M where E is a topological space such that, for all m M : (i) the fiber Em := 1 ({m}) is equipped with the structure of a vector space over R. (ii) There is an neighborhood U M of m and a local trivialization : 1 (U) U Rk which, for each x U, maps the fiber E x to {x} Rk by a linear isomorphism.

FALL 2012 MATH 8230 (VECTOR BUNDLES) LECTURE NOTES 1. DEFINITIONS: VECTOR BUNDLES AND STRUCTURE GROUPS A vector bundle over a topological space M (or “with base space M”) is, essentially, family of vector spaces continuously parametrized by M. (I’m using the letter M to denote the base space of the vector bundle as a concession to the fact that in …

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