# nLab short exact sequence of vector bundles

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Definition

Give a topological ground field $k$ and a base topological space $B$, a short exact sequence of topological $k$-vector bundles over $B$ is a sequence of topological $k$-vector bundle homomorphisms over $B$ (i.e. continuous functions which are fiber-wise $k$-linear maps)

(1)$0 \to \mathcal{V}_L \overset{ \;\; i \;\;}{\longrightarrow} \mathcal{V} \overset{ \;\; p \;\; }{\longrightarrow} \mathcal{V}_R \to 0 \;\;\;\; \in k VectorBundles_B$

(where $0$ denotes the rank-zero bundle) such that $p$ is a surjection and $i = ker_B(p)$ in the injection of its fiber-wise kernel, hence such that over each point $b \colon \ast \overset{}{\longrightarrow} B$ we have a short exact sequence of $k$-vector spaces:

$\underset{ b \in B }{\forall} \;\;\; 0 \to b^\ast \mathcal{V}_L \overset{ \;\; b^\ast i \;\; }{\longrightarrow} b^\ast \mathcal{V} \overset{ \;\; b^\ast p \;\; }{\longrightarrow} b^\ast \mathcal{V}_R \to 0 \;\;\; \in \; k VectorSpaces \,.$

## Properties

### Splitting

###### Proposition

If

1. the ground field is the real numbers $k = \mathbb{R}$,

2. the base space $B$ is a paracompact Hausdorff space,

3. the ranks are all finite,

then every short exact sequence of topological vector bundles (1) splits and exhibits the middle item as the direct sum of vector bundles, over $B$, of the left and the right item:

$\mathcal{V} \;\simeq\; \mathcal{V}_L \oplus_B \mathcal{V}_R \,.$

(e.g. Hatcher, Prop. 1.3, Freed, Lemma 5.6)

###### Proof

Sketch: Under the assumption on $B$, there exists (by this Prop.) a fiberwise inner product on $\mathcal{V}$. With this the splitting follows by th usual splitting of short exact sequences of real vector spaces, applied fiberwise: $\mathcal{V}_R$ is fiberwise identified with the orthogonal complement of $\mathcal{V}_L$.

Textbook accounts:

Lecture notes: