Topology/Vector Bundles

A vector bundle is, broadly speaking, a family of vector spaces which is continuously indexed by a topological space. An important example is the tangent bundle of a manifold.

The formal definition is the following:

Definition (Vector bundle) A real vector bundle on a topological space $B$ is a space $E$ together with a continuous map $p:E\rightarrow B$ with the following properties:

(1) For each $b\in B$, $p^{-1}(b)$ is isomorphic to $\mathbf {R} ^{n}$

(2) $B$ is covered by open sets $U_{i}$ such that there exist homeomorphisms $h_{i}:p^{-1}(U_{i})\rightarrow U_{i}\times \mathbf {R} ^{n}$ and $h_{i}\circ h_{j}^{-1}U_{i}\cap U_{j}\times \mathbf {R} ^{n}\rightarrow U_{i}\cap U_{j}\times \mathbf {R} ^{n}$ is the identity on the first factor and a linear isomorphism on the second.

Replacing $\mathbf {R}$ with $\mathbf {C}$, we get the definition of a complex vector bundle.

We call $E$ the total space of the vector bundle and $B$, the base space.

One can define a smooth vector bundle as following:

$E$ and $B$ have to be smooth manifolds and every maps appearing in the previous definition have to be smooth.

As we have stated before, the tangent bundle of a smooth manifold is a smooth vector bundle.