Topological real vector bundle

Let $X$ be a topological space and let $r\ge 0$ be an integer. A topological real vector bundle of rank $r$ over $X$ is a topological space $E$ , a continuous surjection $\pi:E\to X$ , and a real vector-space structure on every fiber $E_x=\pi^{-1}(x)$ , such that every $x\in X$ has an open neighborhood $U$ and a homeomorphism

\varphi:\pi^{-1}(U)\xrightarrow{\cong} U\times\mathbb R^r

with $\operatorname{pr}_1\circ\varphi=\pi$ and such that each restricted map

\varphi_x:E_x\longrightarrow\{x\}\times\mathbb R^r

is a real-linear isomorphism. The bundle has constant rank $r$ ; its fibers are real vector spaces and its transition maps are real-linear. No smooth structure on $X$ or $E$ is part of this definition.

Bundle maps and pullbacks

A bundle map $E\to F$ over $X$ is continuous and restricts on each fiber to a real-linear map. A continuous map $f:Y\to X$ pulls $E$ back to the fiber product

f^*E=\{(y,e)\in Y\times E:f(y)=\pi(e)\}\longrightarrow Y,

which is again a topological real vector bundle of rank $r$ .

Examples

The product $X\times\mathbb R^r\to X$ is the trivial rank- $r$ bundle. A real line bundle is the rank-one case. The Möbius bundle is a nontrivial real line bundle over the circle. A smooth real vector bundle on a smooth manifold has an underlying topological real vector bundle.

Direct sums and complexification

For two bundles on the same base, $(E\oplus F)_x=E_x\oplus F_x$ . Simultaneous local trivializations give charts $(E\oplus F)|_U\cong U\times\mathbb R^{r+s}$ , which define its topology. The complexification has fibers $E_x\otimes_{\mathbb R}\mathbb C$ ; its charts use the same real transition matrices regarded as complex matrices. Thus $E^{\mathbb C}$ is a topological complex vector bundle of complex rank $r$ .