nLab short exact sequence of vector bundles

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

(over paracompact topological spaces short exact sequences of real vector bundles split)

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 $\mathcal{V}_L$.

Related concepts

• vector bundle

  • real vector bundle

  • complex vector bundle

    • holomorphic vector bundle, pseudoholomorphic vector bundle

  • universal vector bundle

  • dual vector bundle

  • module bundle

  • direct sum of vector bundles

  • connection on a vector bundle

  • flat vector bundle

  • real vector bundle, complex vector bundle

  • super vector bundle

References

Textbook accounts:

• Allen Hatcher, Prop. 1.3 in: Vector bundles and K-Theory (webpage, pdf)

Lecture notes:

• Daniel Freed, More on stabilization, Lecture 5 in Bordism: Old and New (pdf)