Topological complex vector bundle

Let $X$ be a topological space and let $r\geq 0$ . A topological complex vector bundle of rank $r$ over $X$ is a topological space $E$ , a continuous surjection $\pi:E\to X$ , and a complex 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 C^r

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

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

is a complex-linear isomorphism. The bundle has constant rank $r$ ; the fibers and the transition maps are complex-linear.

A bundle map $E\to F$ over $X$ is continuous and restricts on each fiber to a complex-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)\}\to Y,

which is again a topological complex vector bundle of rank $r$ . Pullback is the operation used in the naturality axiom for integral Chern classes.

Direct sums

For bundles $E\to X$ and $F\to X$ , their direct sum is the topological complex vector bundle

E\oplus F=\coprod_{x\in X}(E_x\oplus_{\mathbb C}F_x)\to X.

On trivializing neighborhoods for both bundles, the product trivialization identifies this with $U\times\mathbb C^{r+s}$ , using the standard complex-linear isomorphism $\mathbb C^r\oplus\mathbb C^s\cong\mathbb C^{r+s}$ . For smooth bundles on a smooth manifold this agrees with the corresponding smooth direct sum.

Examples

The product $X\times\mathbb C^r\to X$ is the trivial rank- $r$ bundle. A complex line bundle is the rank-one case. The tautological line bundle $\mathcal O(-1)\to\mathbb{CP}^n$ has fiber the line represented by a point $[z]\in\mathbb{CP}^n$ , and is a basic normalization example for integral Chern classes.

Scope

No smooth structure on $X$ or $E$ is part of this definition. A smooth complex vector bundle on a smooth manifold has an underlying topological complex vector bundle, but the topological definition applies to arbitrary bases.