Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space ${\displaystyle X}$ (for example ${\displaystyle X}$ could be a topological space, a manifold, or an algebraic variety): to every point ${\displaystyle x}$ of the space ${\displaystyle X}$ we associate (or "attach") a vector space ${\displaystyle V(x)}$ in such a way that these vector spaces fit together to form another space of the same kind as ${\displaystyle X}$ (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over ${\displaystyle X}$.

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space ${\displaystyle V}$ such that ${\displaystyle V(x)=V}$ for all ${\displaystyle x}$ in ${\displaystyle X}$: in this case there is a copy of ${\displaystyle V}$ for each ${\displaystyle x}$ in ${\displaystyle X}$ and these copies fit together to form the vector bundle ${\displaystyle X\times V}$ over ${\displaystyle X}$. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.

Vector bundles are almost always required to be locally trivial, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.