nLab
direct sum of vector bundles

Context

Bundles

Linear algebra

Contents

Idea

For E 1,E 2XE_1, E_2 \to X two vector bundles, their direct sum over XX, also called their Whitney sum, is the vector bundle E 1E 2XE_1 \oplus E_2 \to X whose fiber over any xXx \in X is the direct sum of vector spaces of the fibers of E 1E_1 and E 2E_2.

Definition

Definition

(direct sum of topological vector bundles via total spaces)

Let

  1. XX be a topological space,

  2. E 1π 1XE_1 \overset{\pi_1}{\to} X and E 2π 2XE_2 \overset{\pi_2}{\to} X two topological vector bundles over XX.

Then the direct sum of vector bundles E 1 XE 2EE_1 \oplus_X E_2 \to E is the topological vector bundle whose total space is the topological subspace

E 1 XE 2{(v 1,v 2)E 1×E 2|π 1(v 1)=π 2(v 2)}E 1×E 2 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

E 1 XE 2 AAπAA X (v 1,v 2) AAA π 1(v 1)=π 2(v 2). \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 xXx \in X the vector space structure on the fibers

(E 1E 2) x(E 1) x(E 2) x (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 XX be a topological space, and let E 1XE_1 \to X and E 2XE_2 \to X be two topological vector bundles over XX.

Let {U iX} iI\{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

{(g 1) ij:U iU jGL(n 1)}AAAandAAA{(g 2) ij:U iU jGL(n 2)} \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,jIi, j \in I write

(g i) ij(g 2) ij : U iU j GL(n 1+n 2) x AAA ((g 1) ij(x) 0 0 (g 2) ij(x)) \array{ (g_i)_{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 1E 2E_1 \oplus E_2 is the one glued from this direct sum of the transition functions (by this construction):

E 1E 2((iU i)×( n 1+n 2))/({(g 1) ij(g 2) ij} i,jI). 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 XX and YY be topological spaces, and write XYX \sqcup Y for their disjoint union space.

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

More explicitiy: let

i X:Vect(X)Vect(XY) i_X \colon Vect(X) \longrightarrow Vect(X \sqcup Y)

and

i Y:Vect(Y)Vect(X×Y) 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)×Vect(Y)i X (XY)i YVect(XY) 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. XX be a paracompact Hausdorff space,

  2. EXE \to X a topological vector bundle.

Then every vector subbundle E 1EE_1 \hookrightarrow E is a direct vector bundle summand, in that there exists another vector subbundle E 2EE_2 \hookrightarrow E such that their direct sum of vector bundles (def. 1) is EE

E 1E 2E. E_1 \oplus E_2 \simeq E \,.

(e.g. Hatcher, prop. 1.3)

Proof

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

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

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

EE X(E) . E \;\simeq\; E' \oplus_X (E')^\perp \,.
Proposition

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

Let

  1. XX be a compact Hausdorff space;

  2. EXE \to X a topological vector bundle.

Then there exists another topological vector bundle E˜X\tilde E \to X such that the direct sum of vector bundles (def. 1) of the two is a trivial vector X× nX \times \mathbb{R}^n:

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

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

Proof

Let {U iX} iI\{U_i \subset X\}_{i \in I} be an open cover of XX over which EXE \to X has a local trivialization

{ϕ i:U i× nE| U i} iI. \left\{ \phi_i \;\colon\; U_i \times \mathbb{R}^n \overset{\simeq}{\longrightarrow} E\vert_{U_i} \right\}_{i \in I} \,.

By compactness of XX, there is a finite sub-cover, hence a finite set JIJ \subset I such tat

{U iX} iJI \{U_i \subset X\}_{i \in J \subset I}

is still an open cover over which EE trivializes.

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

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

with support supp(f i)U isupp(f_i) \subset U_i. Hence the functions

E| U i AAAA U i× n v AAA f i(x)ϕ i 1(v) \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 iϕ i 1:EX× n. 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

iJf iϕ i:EX×(iJ n)X× n|J|˙. \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 iϕ i 1f_i \cdot \phi^{-1}_i, this is a fiber-wise injective, because at each point at least one of the f if_i is non-vanishing. Hence this is an injection of EE into a trivial vector bundle.

With this the statement follows by prop. 1.

Remark

Prop. 2 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

Revised on May 31, 2017 15:09:17 by Urs Schreiber (46.183.103.17)