nLab direct sum of vector bundles

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 $E_2$ (the fiber-wise direct sum).

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

Whitney summands of trivial vector bundles

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 $E$

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

(e.g. Hatcher, prop. 1.3)

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 $X \times \mathbb{R}^n$:

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

(e.g. Hatcher, prop. 1.4, Friedlander, ptop. 3.1)

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.

Since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity there exists a partition of unity

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

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.

Remark : Let $E_1\rightarrow M$ and $E_2\rightarrow M$ be vector bundles over $M$. This gives product map $E_1\times E_2\rightarrow M\times M$ which is still a vector bundle. Consider diagonal map $d:M\rightarrow M\times M$ given by $m\mapsto (m,m)$. The Whitney sum of $E_1\rightarrow M$ and $E_2\rightarrow M$ is the pull back of $E_1\times E_2\rightarrow M\times M$ along the diagonal map $d:M\rightarrow M\times M$ which is denoted by $E_1\oplus E_2\rightarrow M$.

Characteristic classes of Whitney sums

Proposition

(Euler class takes Whitney sum to cup product)

The Euler class of the Whitney sum of two oriented real vector bundles to the cup product of the separate Euler classes:

$$\chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,.$$

For details see at Euler class, this Prop..

Related concepts

• external direct sum of vector bundles

• inner product on vector bundles

• tensor product of vector bundles, external tensor product of vector bundles, dual vector bundle,

• tensor category

• Stiefel-Whitney class

• virtual vector bundle, topological K-theory

• framed manifold, Thom spectrum

References

Discussion in a context of topological K-theory:

• Max Karoubi, §4.8 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226 Springer (1978) [pdf, doi:10.1007%2F978-3-540-79890-3]

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

and with an eye towards algebraic K-theory;

• Eric Friedlander, An introduction to K-theory (emphasis on algebraic K-theory), 2007 (pdf)