Splitting theorem for vector bundles

Statement

Let $X$ be paracompact Hausdorff, and let

0\longrightarrow E'\overset{\iota}{\longrightarrow}E \overset{q}{\longrightarrow}E''\longrightarrow0

be a short exact sequence of finite-rank real or complex topological vector bundles over $X$ . The splitting theorem for vector bundles states that there is a continuous fiberwise linear bundle map $s:E''\to E$ satisfying $q\circ s=\operatorname{id}_{E''}$ . Equivalently,

E\cong E'\oplus E''

through an isomorphism that identifies $\iota$ with inclusion of the first summand and $q$ with projection onto the second. Here the direct sum uses fiberwise direct sums with their bundle topology. For a smooth exact sequence over a smooth manifold, the splitting and isomorphism may be chosen smooth. A splitting exists, but the theorem does not select a canonical one.

Proof idea

Paracompactness provides a continuous positive-definite fiber metric on $E$ (a Hermitian metric in the complex case), which may be chosen smooth for smooth bundles. The image $\iota(E')$ is a subbundle, and its fiberwise orthogonal complement $\iota(E')^\perp$ is another subbundle. The restriction

q|_{\iota(E')^\perp}:\iota(E')^\perp\longrightarrow E''

is a fiberwise isomorphism and hence a bundle isomorphism in the chosen topological or smooth category. Its inverse, followed by the inclusion into $E$ , is the required right inverse $s$ .

Different bundle metrics generally produce different complements. The resulting direct-sum decompositions are therefore auxiliary choices rather than additional structure carried by the original exact sequence.

Consequences and limits

Every vector subbundle $F\subseteq E$ over a paracompact base has a complementary subbundle $F^\perp$ with $E\cong F\oplus F^\perp$ . On sections, a chosen splitting decomposes each section of $E$ into components in the two summands.

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: chapter 3, bundle metrics, complements, and exact sequences of vector bundles.
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. DOI record. Relevant: chapter 10, vector bundles, bundle metrics, and orthogonal complements.