# nLab direct sum of vector bundles

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Idea

For $E_1, E_2 \to X$ two vector bundles, their direct sum over $X$, also called their Whitney sum, is the vector bundle $E_1 \oplus E_2 \to X$ whose fiber over any $x \in X$ is the direct sum of vector spaces of the fibers of $E_1$ and $E_2$.

## Definition

###### Definition

(direct sum of topological vector bundles via total spaces)

Let

1. $X$ be a topological space,

2. $E_1 \overset{\pi_1}{\to} X$ and $E_2 \overset{\pi_2}{\to} X$ two topological vector bundles over $X$.

Then the direct sum of vector bundles $E_1 \oplus_X E_2 \to E$ is the topological vector bundle whose total space is the topological subspace

$E_1 \oplus_X E_2 \;\coloneqq\; \left\{ (v_1, v_2) \in E_1 \times E_2 \,\vert\, \pi_1(v_1) = \pi_2(v_2) \right\} \;\subset\; E_1 \times E_2$

of the product topological space of the two total spaces, and whose projection map is

$\array{ E_1 \oplus_X E_2 &\overset{\phantom{AA}\pi\phantom{AA}}{\longrightarrow}& X \\ (v_1,v_2) &\overset{\phantom{AAA}}{\mapsto}& \pi_1(v_1) = \pi_2(v_2) } \,.$

For $x \in X$ the vector space structure on the fibers

$(E_1 \oplus E_2)_x \simeq (E_1)_x \oplus (E_2)_x$

is the one on the direct sum of vector spaces.

###### Definition

(direct sum of topological vector bundles via transition functions)

Let $X$ be a topological space, and let $E_1 \to X$ and $E_2 \to X$ be two topological vector bundles over $X$.

Let $\{U_i \subset X\}_{i \in I}$ be an open cover with respect to which both vector bundles locally trivialize (this always exists: pick a local trivialization of either bundle and form the joint refinement of the respective open covers by intersection of their patches). Let

$\left\{ (g_1)_{i j} \colon U_i \cap U_j \to GL(n_1) \right\} \phantom{AAA} \text{and} \phantom{AAA} \left\{ (g_2)_{i j} \colon U_i \cap U_j \longrightarrow GL(n_2) \right\}$

be the transition functions of these two bundles with respect to this cover.

For $i, j \in I$ write

$\array{ (g_1)_{i j} \oplus (g_2)_{i j} &\colon& U_i \cap U_j &\longrightarrow& GL(n_1 + n_2) \\ && x &\overset{\phantom{AAA}}{\mapsto}& \left( \array{ (g_1)_{i j}(x) & 0 \\ 0 & (g_2)_{i j}(x) } \right) }$

be the pointwise direct sum of these transition functions

Then the direct sum bundle $E_1 \oplus E_2$ is the one glued from this direct sum of the transition functions (by this construction):

$E_1 \oplus E_2 \;\coloneqq\; \left( \left( \underset{i}{\sqcup} U_i \right) \times \left( \mathbb{R}^{n_1 + n_2} \right) \right)/ \left( \left\{ (g_1)_{i j} \oplus (g_2)_{i j} \right\}_{i,j \in I} \right) \,.$

## Examples

###### Example

Let $X$ and $Y$ be topological spaces, and write $X \sqcup Y$ for their disjoint union space.

Then every topological vector bundle on $X \sqcup Y$ is the direct sum of a vector bundle that has rank zero on $Y$ and one that has rank zero on $X$.

More explicitiy: let

$i_X \colon Vect(X) \longrightarrow Vect(X \sqcup Y)$

and

$i_Y \colon Vect(Y) \longrightarrow Vect(X \times Y)$

be the operations of extending a vector bundle on the other connected component by a rank-0 vector bundle, then

$Vect(X) \times Vect(Y) \underoverset{\simeq}{ i_X \oplus_{(X \sqcup Y)} i_Y }{\longrightarrow} Vect(X \sqcup Y)$

is an isomorphism of isomorphism classes of vector bundles (and an equivalence of categories of categories of vector bundles before passing to isomorphism classes).

## Properties

###### Proposition

(sub-bundles over paracompact spaces are direct summands)

Let

1. $X$ be a paracompact Hausdorff space,

2. $E \to X$ a topological vector bundle.

Then every vector subbundle $E_1 \hookrightarrow E$ is a direct vector bundle summand, in that there exists another vector subbundle $E_2 \hookrightarrow E$ such that their direct sum of vector bundles (def. ) is $E$

$E_1 \oplus E_2 \simeq E \,.$
###### Proof

Since $X$ is assumed to be paracompact Hausdorff, there exists a inner product on vector bundles

$\langle -,-\rangle \;\colon\; E \oplus_X E \longrightarrow X \times \mathbb{R}$

(by this prop.). This defines at each $x \in X$ the orthogonal complement $(E'_x)^\perp \subset E_x$ of $E'_x \hookrightarrow E$. The subspace of these orthogonal complements is readily checked to be a topological vector bundle $(E')^\perp \to X$. Hence by construction we have

$E \;\simeq\; E' \oplus_X (E')^\perp \,.$
###### Proposition

(over compact Hausdorff spaces every vector bundle is direct summand of a trivial bundle)

Let

1. $X$ be a compact Hausdorff space;

2. $E \to X$ a topological vector bundle.

Then there exists another topological vector bundle $\tilde E \to X$ such that the direct sum of vector bundles (def. ) of the two is a trivial vector $X \times \mathbb{R}^n$:

$E \oplus \tilde E \;\simeq\; X \times \mathbb{R}^n \,.$
###### Proof

Let $\{U_i \subset X\}_{i \in I}$ be an open cover of $X$ over which $E \to X$ has a local trivialization

$\left\{ \phi_i \;\colon\; U_i \times \mathbb{R}^n \overset{\simeq}{\longrightarrow} E\vert_{U_i} \right\}_{i \in I} \,.$

By compactness of $X$, there is a finite sub-cover, hence a finite set $J \subset I$ such tat

$\{U_i \subset X\}_{i \in J \subset I}$

is still an open cover over which $E$ trivializes.

$\left\{ f_i \;\colon\; X \to [0,1] \right\}_{i \in J}$

with support $supp(f_i) \subset U_i$. Hence the functions

$\array{ E\vert_{U_i} &\overset{\phantom{AAAA}}{\longrightarrow}& U_i \times \mathbb{R}^n \\ v &\overset{\phantom{AAA}}{\mapsto}& f_i(x) \cdot \phi_i^{-1}(v) }$

extend by 0 to vector bundle homomorphism of the form

$f_i \cdot \phi^{-1}_i \;\colon\; E \longrightarrow X \times \mathbb{R}^n \,.$

The finite pointwise direct sum of these yields a vector bundle homomorphism of the form

$\underset{i \in J}{\oplus} f_i \cdot \phi_i \;\colon\; E \longrightarrow X \times \left( \underset{i \in J}{\oplus} \mathbb{R}^n \right) \simeq X \times \mathbb{R}^{n \dot {\vert J\vert}} \,.$

Observe that, as opposed to the single $f_i \cdot \phi^{-1}_i$, this is a fiber-wise injective, because at each point at least one of the $f_i$ is non-vanishing. Hence this is an injection of $E$ into a trivial vector bundle.

With this the statement follows by prop. .

###### Remark

Prop. is key for the construction of topological K-theory groups on compact Hausdorff spaces.

## References

Discussion with an eye towards topological K-theory is in

• Max Karoubi, K-theory. An introduction, Grundlehren der Mathematischen Wissenschaften 226, Springer 1978. xviii+308 pp.

• Allen Hatcher, section 1.1 of Vector bundles and K-Theory, (partly finished book) web

and with an eye towards algebraic K-theory in

Last revised on November 2, 2018 at 01:31:35. See the history of this page for a list of all contributions to it.