nLab
short exact sequence of vector bundles

Contents

Context

Bundles

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

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

(1)0𝒱 Li𝒱p𝒱 R0kVectorBundles B 0 \to \mathcal{V}_L \overset{ \;\; i \;\;}{\longrightarrow} \mathcal{V} \overset{ \;\; p \;\; }{\longrightarrow} \mathcal{V}_R \to 0 \;\;\;\; \in k VectorBundles_B

(where 00 denotes the rank-zero bundle) such that pp is a surjection and i=ker B(p)i = ker_B(p) in the injection of its fiber-wise kernel, hence such that over each point b:*Bb \colon \ast \overset{}{\longrightarrow} B we have a short exact sequence of kk-vector spaces:

bB0b *𝒱 Lb *ib *𝒱b *pb *𝒱 R0kVectorSpaces. \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=k = \mathbb{R},

  2. the base space BB 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 BB, of the left and the right item:

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

References

Textbook accounts:

Lecture notes:

Last revised on August 16, 2021 at 11:23:45. See the history of this page for a list of all contributions to it.