nLab topological vector bundle

Idea

A topological vector bundle is a vector bundle in the context of topology: a continuously varying collection of vector space over a given topological space.

For more survey and motivation see at vector bundle. Here we discuss the details of the general concept in topology. See also differentiable vector bundle and algebraic vector bundle.

Definition

We first give the more abstract definiton in terms of slice categories (def. below) and then unwind this to the traditional definition (def below).

In the following

• $k$ is either the topological field

  • $k = \mathbb{R}$ of real numbers

  • or $k = \mathbb{C}$ of complex numbers

  equipped with the Euclidean metric topology.

• vector space means finite dimensional vector space.

In terms of slice categories

Definition

(topological vector bundles in terms of slice categories)

Write Top for the category of topological spaces, and for $X \in Top$ a space, write $Top_{/X}$ for its slice category over $X$. The Cartesian product in $Top_{/X}$ is the fiber product over $X$ in $Top$, which we denote by $(-) \times_X (-)$. Observe $[X \times k \to X] \in Top_{/X}$ is canonically a field internal to $Top_{/X}$

A topological vector bundle over $X \in Top$ is

1. an object $[E \overset{\pi}{\to} X]$ of $Top_{/X}$

2. with the structure of an $X \times k$-vector space-object internal to $Top_{/X}$, hence

   1. a morphism $(-)+(-) \;\colon\; E \times_X E \to E$

   2. a morphism $(-)\cdot(-) \;\colon\; k \times E \to E$

   which satisfy the vector space axioms

such that

• (local triviality) there exists

  1. an open cover $\{U_i \subset X\}_{i \in I}$, regarded via the disjoint union space $U \coloneqq \underset{i \in I}{\sqcup} U_i$ of the patches as the object $[U \to X] \in Top_{/X}$,

  2. an isomorphism of vector space objects in $Top_{/U}$

     $$U \times \mathbb{R}^n \overset{\simeq}{\longrightarrow} U \times_X E \,,$$

     for some $n \in \mathbb{N}$, where $[U \times k^n \overset{pr_1}{\to} X]$ and $[U \times_X E \overset{pr_1}{\to} U]$ are regarded as a vector space objects in $Top_{U}$ in the canonical way.

It follows that $n \in \mathbb{N}$ is constant on connected components. Often this is required to be constant on all of $X$ and then called the rank of the vector bundle.

A homomorphism of topological vector bundles is simple a homomorphism of vector space objects in $Top_{/X}$.

Topological vector bundles over $X$ and homomorphisms between them constitutes a category, usually denoted Vect(X).

Notice that viewed in Top, the last condition means that there is a diagram of the form

$$\array{ U \times k^n &\overset{\simeq}{\longrightarrow}& U \times_X E &\overset{}{\longrightarrow}& E \\ & \searrow & \downarrow &(pb)& \downarrow^{\mathrlap{\pi}} \\ && U &\longrightarrow& X }$$

where the square is a pullback square and the homeomorphism in the top left is fiber-wise linear.

If we say this yet more explicitly, it yields the definition as found in the traditional textbooks:

In components

Definition

(topological vector bundle in components)

Let $X$ be a topological space. Then a topological vector bundle over $X$ is

1. a topological space $E$;

2. a continuous function $E \overset{\pi}{\to} X$

3. for each $x \in X$ the stucture of a finite-dimensional $k$-vector space on the pre-image

   $$E_x \coloneqq \pi^{-1}(\{x\}) \subset E$$

such that this is locally trivial in that there exists

1. an open cover $\{U_i \subset X\}_{i \in I}$,

2. for each $i \in I$ an $n_i \in \mathbb{N}$ and a homeomorphism

   $$\phi_i \;\colon\; U_i \times k^{n_i} \overset{\simeq}{\longrightarrow} \pi^{-1}(U_i) \subset E$$

   from the product topological space of $U_i$ with the real numbers (equipped with their Euclidean space metric topology) to the restriction of $E$ over $U_i$, such that

   1. $\phi_i$ is a map over $U_i$ in that $\pi \circ \phi_i = pr_1$, hence in that $\phi_i(\{x\} \times k^{n_i}) \subset \pi^{-1}(\{x\})$

   2. $\phi_i$ is a linear map in each fiber in that

      $$\underset{x \in U_i}{\forall} \left( \phi_i(x) \;\colon\; k^{n_i} \overset{\text{linear}}{\longrightarrow} E_x = \pi^{-1}(\{x\}) \right) \,.$$

Often, but not always, it is required that the numbers $n_i$ are all equal to some $n \in \mathbb{N}$, for all $i \in I$, hence that the vector space fibers all have the same dimension. In this case one says that the vector bundle has rank $n$. (Over a connected topological space this is automatic, but the fiber dimension may be distinct over distinct connected components.)

For $[E_1 \overset{\pi_1}{\to} X]$ and $[E_2 \overset{\pi_2}{\to} X]$ two topological vector bundles over the same base space, then a homomorphism between them is

• a continuous function $f \colon E_1 \longrightarrow E_2$

such that

1. $f$ respects the projections: $\pi_2 \circ f = \pi_1$;

2. for each $x \in X$ we have that $f|_x \colon (E_1)_x \to (E_2)_x$ is a linear map.

Remark

(category of topological vector bundles)

For $X$ a topological space, there is the category whose

• objects are the topological vector bundles over $X$,

• morphisms are the topological vector bundle homomorphisms

according to def. . This category usually denoted Vect(X).

We write $Vect(X)_{/\sim}$ for the set of isomorphism classes of this category.

Remark

(fiberwise operations)
The category FinDimVect of finite dimensional vector spaces over a topological ground field is canonically a Top-enriched category, and so are hence its product categories $FinDimVect^{n}$, for $n \in \mathbb{N}$. Any Top-enriched functor

$$F \;\colon\; FinDimVect^n \longrightarrow FinDimVect$$

induces a functorial construction of new topological vector bundles $\widehat{F}(\mathcal{V}_1,, \cdots, \mathcal{V}_n)$ from any n-tuple $(\mathcal{V}_1, \mathcal{V}_2 , \cdots, \mathcal{V}_n)$ of vector bundles over the same base space $B$, by taking the new fiber over a point $b \in B$ to be (e.g. Milnor & Stasheff 1974, p. 32):

$$F \big( \mathcal{V}_1, \cdots \mathcal{V}_n \big) _b \;\coloneqq\; F \big( (\mathcal{V}_1)_b, \cdots, (\mathcal{V}_n)_b \big) \,.$$

For example:

• if $F \,\coloneqq\, (-)^\ast \,\colon\, FinDimVect \longrightarrow FinDimVect$ is the operation of forming dual vector spaces, then $\widehat{F}$ constructs the fiberwise dual vector bundle;

• if $F \,\coloneqq\, det \,\colon\, FinDimVect \longrightarrow FinDimVect$ is the operation of forming determinants, then $\widehat{F}$ is the construction of fiberwise determinant line bundles;

• if $F \,\coloneqq\, \oplus \,\colon\, FinDimVect^2 \longrightarrow FinDimVect$ is the direct sum of vector space, then $\widehat{F}$ constructs the fiberwise direct sum of vector bundles (“Whitney sum”);

• if $F \,\coloneqq\, \otimes \,\colon\, FinDimVect^2 \longrightarrow FinDimVect$ is the tensor product of vector spaces, then $\widehat{F}$ constructs the fiberwise tensor product of vector bundles.

Remark

(some terminology)

Let $k$ and $n$ be as in def. . Then:

For $k = \mathbb{R}$ one speaks of real vector bundles.

For $k = \mathbb{C}$ one speaks of complex vector bundles.

For $n = 1$ one speaks of line bundles, in particular of real line bundles and of complex line bundles.

Remark

(any two topologial vector bundles have local trivialization over a common open cover)

Let $[E_1 \to X]$ and $[E_2 \to X]$ be two topological vector bundles (def. ). Then there always exists an open cover $\{U_i \subset X\}_{i \in I}$ such that both bundles have a local trivialization over this cover.

Proof

By definition we may find two possibly different open covers $\{U^1_{i_1} \subset X\}_{{i_1} \in I_1}$ and $\{U^2_{i_2} \subset X\}_{i_2 \in I_2}$ with local tivializations $\{ U^1_{i_1} \underoverset{\simeq}{\phi^1_{i_1}}{\to} E_1\vert_{U^1_{i_1}} \}_{i_1 \in I_1}$ and $\{ U^2_{i_2} \underoverset{\simeq}{\phi^2_{i_2}}{\to} E_2\vert_{U^2_{i_2}} \}_{i_2 \in I_2}$.

The joint refinement of these two covers is the open cover

$$\left\{ U_{i_1, i_2} \coloneqq U^1_{i_1} \cap U^2_{i_2} \subset X \right\}_{(i_1, i_2) \in I_1 \times I_2} \,.$$

The original local trivializations restrict to local trivializations on this finer cover

$$\left\{ U_{i_1, i_2} \underoverset{\simeq}{\phi^1_{i_1}\vert_{U^2_{i_2}}}{\longrightarrow} E_1\vert_{U_{i_1, i_2}} \right\}_{(i_1, i_2) \in I_1 \times I_2}$$

and

$$\left\{ U_{i_1, i_2} \underoverset{\simeq}{\phi^2_{i_2}\vert_{U^1_{i_1}}}{\longrightarrow} E_2\vert_{U_{i_1, i_2}} \right\}_{(i_1, i_2) \in I_1 \times I_2} \,.$$

Example

(trivial topological vector bundle and (local) trivialization)

For $X$ any topological space, and $n \in \mathbb{N}$, we have that the product topological space

$$X \times k^n \overset{pr_1}{\to} X$$

canonically becomes a topological vector bundle over $X$ (def. ). This is called the trivial vector bundle of rank $n$ over $X$.

Given any topological vector bundle $E \to X$, then a choice of isomorphism to a trivial bundle (if it exists)

$$E \overset{\simeq}{\longrightarrow} X \times k^n$$

is called a trivialization of $E$. A vector bundle for which a trivialization exists is called trivializable.

Accordingly, the local triviality condition in the definition of topological vector bundles (def. ) says that they are locally isomorphic to the trivial vector bundle. One also says that the data consisting of an open cover $\{U_i \subset X\}_{i \in I}$ and the homeomorphisms

$$\left\{ U_i \times k^n \overset{\simeq}{\to} E|_{U_i} \right\}_{i \in I}$$

as in def. constitute a local trivialization of $E$.

Example

(section of a topological vector bundle)

Let $E \overset{\pi}{\to} X$ be a topological vector bundle (def. ).

Then a homomorphism of vector bundles from the trivial line bundle (example , remark )

$$f \;\colon\; X \times k \longrightarrow E$$

is, by fiberwise linearity, equivalently a continuous function

$$\sigma \;\colon\; X \longrightarrow E$$

such that $\pi \circ \sigma = id_X$;

$$f(x, c) = c \sigma(x)$$

Such functions $\sigma \colon X \to E$ are called sections (or cross-sections) of the vector bundle $E$.

Example

(topological vector sub-bundle)

Given a topological vector bundel $E \to X$ (def. ), then a sub-bundle is a homomorphism of topological vector bundles over $X$

$$i\;\colon\; E' \hookrightarrow E$$

such that for each point $x \in X$ this is a linear embedding of fibers

$$i|_x \;\colon\; E'_x \hookrightarrow E_x \,.$$

(This is a monomorphism in the category $Vect(X)$ of topological vector bundles over $X$.)

Transition functions and Cech cohomology

We discuss how topological vector bundles are equivalently given by cocycles in Cech cohomology constituted by their transition functions.

Definition

(continuous functions on open subsets with values in the general linear group)

For $n \in \mathbb{N}$, regard the general linear group $GL(n,k)$ as a topological group with its standard topology, given as the Euclidean subspace topology via $GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}$ or as the or as the subspace topology $GL(n,k) \subset Maps(k^n, k^n)$ of the compact-open topology on the mapping space. (That these topologies coincide is the statement of this prop..

For $X$ a topological space, we write

$$\underline{GL(n,k)} \;\colon\; U \mapsto Hom_{Top}(U, GL(n,k) )$$

for the assignment that sends an open subset $U \subset X$ to the set of continuous functions $g \colon U \to GL(n,k)$ (for $U \subset X$ equipped with its subspace topology), regarded as a group via the pointwise group operation in $GL(n,k)$:

$$g_1 \cdot g_2 \;\colon\; x \mapsto g_1(x) \cdot g_2(x) \,.$$

Moreover, for $U' \subset U \subset X$ an inclusion of open subsets, and for $g \in \underline{GL(n,k)}(U)$, we write

$$g|_{U'} \in \underline{GL(n,k)}(U')$$

for the restriction of the continuous function from $U$ to $U'$.

Remark

(sheaf of groups)

In the language of category theory the assignment $\underline{GL(n,k)}$ from def. of continuous functions to open subsets and the restriction operations between these is called a sheaf of groups on the site of open subsets of $X$.

Definition

(transition functions)

Given a topological vector bundle $E \to X$ as in def. and a choice of local trivialization $\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_i}\}$ (example ) there are for $i,j \in I$ induced continuous functions

$$\left\{ g_{i j} \;\colon\; (U_i \cap U_j) \longrightarrow GL(n, k) \right\}_{i,j \in I}$$

to the general linear group (as in def. ) given by composing the local trivialization isomorphisms:

$$\array{ (U_i \cap U_j) \times k^n &\overset{ \phi_i|_{U_i \cap U_j} }{\longrightarrow}& E|_{U_i \cap U_j} &\overset{ \phi_j^{-1}\vert_{U_i \cap U_j} }{\longrightarrow}& (U_i \cap U_j) \times k^n \\ (x,v) && \overset{\phantom{AAA}}{\mapsto} && \left( x, g_{i j}(x)(v) \right) } \,.$$

These are called the transition functions for the given local trivialization.

These functions satisfy a special property:

Definition

(Cech cocycles)

Let $X$ be a topological space.

A normalized Cech cocycle of degree 1 with coefficients in $\underline{GL(n,k)}$ (def. ) is

1. an open cover $\{U_i \subset X\}_{i \in I}$

2. for all $i,j \in I$ a continuous function $g_{i j} \colon U_i \cap U_j \to GL(n,k)$ as in def.

such that

1. (normalization) $\underset{i \in I}{\forall}\left( g_{i i} = const_1 \right)$ (the constant function on the neutral element in $GL(n,k)$),

2. (cocycle condition) $\underset{i,j \in I}{\forall}\left( g_{j k} \cdot g_{i j} = g_{i k}\;\;\text{on}\, U_i \cap U_j \cap U_k\right)$.

Write

$$C^1(X, \underline{GL(n,k)} )$$

for the set of all such cocycles for given $n \in \mathbb{N}$ and write

$$C^1( X, \underline{GL}(k) ) \;\coloneqq\; \underset{n \in \mathbb{N}}{\sqcup} C^1(X, \underline{GL(n,k)})$$

for the disjoint union of all these cocycles as $n$ varies.

Example

(transition functions are Cech cocycles)

Let $E \to X$ be a topological vector bundle (def. ) and let $\{U_i \subset X\}_{i \in I}$, $\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_{i}}\}_{i \in I}$ be a local trivialization (example ).

Then the set of induced transition functions $\{g_{i j} \colon U_i \cap U_j \to GL(n)\}$ according to def. is a normalized Cech cocycle on $X$ with coefficients in $\underline{GL(k)}$, according to def. .

Proof

This is immediate from the definition:

$$\begin{aligned} g_{i i }(x) & = \phi_i^{-1} \circ \phi_i(x,-) \\ & = id_{k^n} \end{aligned}$$

and

$$\begin{aligned} g_{j k}(x) \cdot g_{i j}(x) & = \left(\phi_k^{-1} \circ \phi_j\right) \circ \left(\phi_j^{-1}\circ \phi_i\right)(x,-) \\ & = \phi_k^{-1} \circ \phi_i(x,-) \\ & = g_{i k}(x) \end{aligned} \,.$$

Conversely:

Example

(topological vector bundle constructed from a Cech cocycle)

Let $X$ be a topological space and let $c \in C^1(X, \underline{GL(k)})$ a Cech cocycle on $X$ according to def. , with open cover $\{U_i \subset X\}_{i \in I}$ and component functions $\{g_{i j}\}_{i,j \in I}$.

This induces an equivalence relation on the product topological space

$$\left( \underset{i \in I}{\sqcup} U_i \right) \times k^n$$

(of the disjoint union space of the patches $U_i \subset X$ regarded as topological subspaces with the product space $k^n = \underset{\{1,\cdots, n\}}{\prod} k$) given by

$$\big( ((x,i), v) \;\sim\; ((y,j), w) \big) \;\Leftrightarrow\; \left( (x = y) \;\text{and}\; (g_{i j}(x)(v) = w) \right) \,.$$

Write

$$E(c) \;\coloneqq\; \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right)$$

for the resulting quotient topological space. This comes with the evident projection

$$\array{ E(c) &\overset{\phantom{AA}\pi \phantom{AA}}{\longrightarrow}& X \\ [(x,i,),v] &\overset{\phantom{AAA}}{\mapsto}& x }$$

which is a continuous function (by the universal property of the quotient topological space construction, since the corresponding continuous function on the un-quotientd disjoint union space respects the equivalence relation). Moreover, each fiber of this map is identified with $k^n$, and hence canonicaly carries the structure of a vector space.

Finally, the quotient co-projections constitute a local trivialization of this vector bundle over the given open cover.

Therefore $E(c) \to X$ is a topological vector bundle (def. ). We say it is the topological vector bundle glued from the transition functions.

Remark

(bundle glued from Cech cocycle is a coequalizer)

Stated more category theoretically, the constructure of a topological vector bundle from Cech cocycle data in example is a universal construction in topological spaces, namely the coequalizer of the two morphisms

$$i, \mu: \underset{i j}{\sqcup} (U_i \cap U_j) \times V \overset{\to}{\to} \underset{i}{\sqcup} U_i \times V$$

in the category of vector space objects in the slice category $Top/X$. Here the restriction of $i$ to the coproduct summands is induced by inclusion:

$$(U_i \cap U_j) \times V \hookrightarrow U_i \times V \hookrightarrow \underset{i}{\sqcup} U_i \times V$$

and the restriction of $\mu$ to the coproduct summands is via the action of the transition functions:

$$(U_i \cap U_j) \times V \overset{(\langle incl, g_{i j} \rangle) \times V}{\to} U_j \times GL(V) \times V \overset{action}{\to} U_j \times V \hookrightarrow \underset{j}{\sqcup} U_j \times V$$

In fact, extracting transition functions from a vector bundle by def. and constructing a vector bundle from Cech coycle data as above are operations that are inverse to each other, up to isomorphism.

Proposition

(topological vector bundle reconstructed from its transition functions)

Let $[E \overset{\pi}{\to} X]$ be a topological vector bundle (def. ), let $\{U_i \subset X\}_{i \in I}$ be an open cover of the base space, and let $\left\{ U_i \times k^n \underoverset{\simeq}{\phi_i}{\longrightarrow} E|_{U_i} \right\}_{i \in I}$ be a local trivialization.

Write

$$\left\{ g_{i j} \coloneqq \phi_j^{-1}\circ \phi_i \colon U_i \cap U_j \to GL(n,k) \right\}_{i,j \in I}$$

for the corresponding transition functions (def. ). Then there is an isomorphism of vector bundles over $X$

$$\left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) \;\underoverset{\simeq}{(\phi_i)_{i \in I}}{\longrightarrow}\; E$$

from the vector bundle glued from the transition functions according to def. to the original bundle $E$, whose components are the original local trivialization isomorphisms.

Proof

By the universal property of the disjoint union space (coproduct in Top), continuous functions out of them are equivalently sets of continuous functions out of every summand space. Hence the set of local trivializations $\{U_i \times k^n \underoverset{\simeq}{\phi_i}{\to} E|_{U_i} \subset E\}_{i \in I}$ may be collected into a single continuous function

$$\underset{i \in I}{\sqcup} U_i \times k^n \overset{(\phi_i)_{i \in I}}{\longrightarrow } E \,.$$

By construction this function respects the equivalence relation on the disjoint union space given by the transition functions, in that for each $x \in U_i \cap U_j$ we have

$$\phi_i((x,i),v) = \phi_j \circ \phi_j^{-1} \circ \phi_i((x,i),v) = \phi_j \circ ((x,j),g_{i j}(x)(v)) \,.$$

By the universal property of the quotient space coprojection this means that $(\phi_i)_{i \in I}$ uniquely extends to a continuous function on the quotient space such that the following diagram commutes

$$\array{ \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& E \\ \downarrow & \nearrow_{\exists !} \\ \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) } \,.$$

It is clear that this continuous function is a bijection. Hence to show that it is a homeomorphism, it is now sufficient to show that this is an open map (by this prop.).

So let $O$ be an subset in the quotient space which is open. By definition of the quotient topology this means equivalently that its restriction $O_i$ to $U_i \times k^n$ is open for each $i \in I$. Since the $\phi_i$ are homeomorphsms, it follows that the images $\phi_i(O_i) \subset E\vert_{U_ i}$ are open. By the nature of the subspace topology, this means that these images are open also in $E$. Therefore also the union $f(O) = \underset{i \in I}{\cup} \phi_i(O_i)$ is open.

Definition

(coboundary between Cech cocycles )

Let $X$ be a topological space and let $c_1, c_2 \in C^1(X, \underline{GL(k)})$ be two Cech cocycles (def. ), given by

1. $\{U_i \subset X\}_{i \in I}$ and $\{U'_i \subset X\}_{i' \in I'}$ two open covers,

2. $\{g_{i j} \colon U_i \cap U_j \to GL(k,n_)\}_{i,j \in I}$ and $\{g_'_{i',j'} \colon U'_{i'} \cap U'_{j'} \to GL(n',k) \}_{i', j'}$ the corrsponding component functions.

Then a coboundary between these two cocycles is

1. the condition that $n = n'$,

2. an open cover $\{V_\alpha \subset X\}_{\alpha \in A}$,

3. functions$\phi \colon A \to I$ and $\phi' \colon A \to J$ such that $\underset{\alpha \in A}{\forall}\left( \left( V_\alpha \subset U_{\phi(\alpha)} \right) \,\text{and}\, \left( V_\alpha \subset U'_{\phi'(\alpha)} \right) \right)$

4. a set $\{ \kappa_\alpha \colon V_\alpha \to GL(n,k) \}$ of continuous functions as in def.

such that

• $\underset{ \alpha, \beta \in A }{\forall} \left( \kappa_{\beta} \cdot g_{\phi(\alpha) \phi(\beta)} = g'_{\phi'(\alpha) \phi'(\beta)} \cdot \kappa_{\alpha} \,\, \text{on}\,\, V_\alpha \cap V_\beta \right)$,

  hence such that the following diagrams of linear maps commute for all $\alpha, \beta \in A$ and $x \in V_{\alpha} \cap V_\beta$:

  $$\array{ k^n &\overset{ g_{\phi(\alpha) \phi(\beta)}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_{\alpha}(x)} }\downarrow && \downarrow^{\mathrlap{ \kappa_{\beta}(x) }} \\ k^n &\underset{ g'_{\phi'(\alpha) \phi'(\beta)}(x) }{\longrightarrow}& k^n } \,.$$

Say that two Cech cocycles are cohomologous if there exists a coboundary between them.

Example

(refinement of a Cech cocycle is a coboundary)

Let $X$ be a topological space and let $c \in C^1(X, \underline{GL(k)})$ be a Cech cocycle as in def. , with respect to some open cover $\{U_i \subset X\}_{i \in I}$ given by component functions $\{g_{i j}\}_{i,j \in I}$.

Then for $\{V_\alpha \subset X\}_{\alpha \in A}$ a refinement of the given open cover, hence an open cover such that there exists a function $\phi \colon A \to I$ with $\underset{\alpha \in A}{\forall}\left( V\alpha \subset U_{\phi(\alpha)} \right)$, then

$$g'_{ \alpha \beta } \coloneqq g_{\phi(\alpha) \phi(\beta)} \colon V_\alpha \cap V_\beta \longrightarrow GL(n,k)$$

are the components of a Cech cocycle $c'$ which is cohomologous to $c$.

Proposition

(isomorphism of topological vector bundles induces Cech coboundary between their transition functions)

Let $X$ be a topological space, and let $c_1, c_2 \in C^1(X, \underline{GL(n,k)} )$ be two Cech cocycles as in def. .

Every isomorphism of topological vector bundles

$$f \;\colon\; E(c_1) \overset{\simeq}{\longrightarrow} E(c_2)$$

between the vector bundles glued from these cocycles according to def. induces a coboundary between the two cocycles,

$$c_1 \sim c_2 \,,$$

according to def. .

Proof

By example we may assume without restriction that the two Cech cocycles are defined with respect to the same open cover $\{U_i \subset X\}_{i \in I}$ (for if they are not, then both are cohomologous to cocycles on a joint refinement of the original covers and we may argue with these).

Accordingly, by example the two bundles $E(c_1)$ and $E(c_2)$ both have local trivializations of the form

$$\{ U_i \times k^n \underoverset{\simeq}{\phi^1_i}{\longrightarrow} E(c_1)\vert_{U_i}\}$$

and

$$\{ U_i \times k^n \underoverset{\simeq}{\phi^2_i}{\longrightarrow} E(c_2)\vert_{U_i}\}$$

over this cover. Consider then for $i \in I$ the function

$$f_i \coloneqq (\phi_i^2)^{-1}\circ f\vert_{U_i} \circ \phi^1_i \,,$$

hence the unique function making the following diagram commute:

$$\array{ U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& E(c_1)\vert_{U_i} \\ {}^{\mathllap{f_i}}\downarrow && \downarrow^{\mathrlap{ f }} \\ U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& E(c_2)\vert_{U_i} } \,.$$

This induces for all $i,j \in I$ the following composite commuting diagram

$$\array{ (U_i \cap U_j) \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& E(c_1)\vert_{U_i \cap U_j} & \underoverset{\simeq}{(\phi^1_j)^{-1}}{\longrightarrow} & (U_i \cap U_j) \times k^n \\ {}^{\mathllap{f_i}}\downarrow && \downarrow^{\mathrlap{ f }} && \downarrow^{\mathrlap{ f_j }} \\ (U_i \cap U_j) \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& E(c_2)\vert_{U_1 \cap U_2} &\underoverset{(\phi^2_j)^{-1}}{\simeq}{\longrightarrow}& (U_i \cap U_j) \times k^n } \,.$$

By construction, the two horizonal composites of this diagram are pointwise given by the components $g^1_{i j}$ and $g^2_{i j}$of the cocycles $c_1$ and $c_2$, respectively. Hence the commutativity of this diagram is equivalently the commutativity of these diagrams:

$$\array{ k^n &\overset{ g^1_{i j}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{ f_i(x) } }\downarrow && \downarrow^{\mathrlap{ f_j(x) }} \\ k^n &\underset{ g^2_{ i j }(x) }{\longrightarrow}& k^n } \,.$$

for all $i,j \in I$ and $x \in U_i \cap U_j$. By def. this exhibits the required coboundary.

Definition

(Cech cohomology)

Let $X$ be a topological space. The relation $\sim$ on Cech cocycles of being cohomologous (def. ) is an equivalence relation on the set $C^1( X, \underline{GL(k)} )$ of Cech cocycles (def. ).

Write

$$H^1(X, \underline{GL(k)} ) \;\coloneqq\; C^1(X, \underline{GL(k)} )/\sim$$

for the resulting set of equivalence classes. This is called the Cech cohomology of $X$ in degree 1 with coefficients in $\underline{GL(k)}$.

Proposition

(degree-1 Cech cohomology computes topological vector bundles)

Let $X$ be a topological space.

The construction of gluing a topological vector bundle from a Cech cocycle (example ) constitutes a bijection between the degree-1 Cech cohomology of $X$ with coefficients in $GL(n,k)$ (def. ) and the set of isomorphism classes of topological vector bundles on $X$ (def. , remark ):

$$\array{ H^1(X,\underline{GL(k)}) &\overset{\phantom{AA}\simeq \phantom{AA}}{\longrightarrow}& Vect(X)_{/\sim} \\ c &\overset{\phantom{AAA}}{\mapsto}& E(c) } \,.$$

Proof

First we need to see that the function is well defined, hence that if cocycles $c_1, c_2 \in C^1(X,\underline{GL(k)})$ are related by a coboundary, $c_1 \sim c_2$ (def. ), then the vector bundles $E(c_1)$ and $E(c_2)$ are related by an isomorphism.

Let $\{V_\alpha \subset X\}_{\alpha \in A}$ be the open cover with respect to which the coboundary $\{\kappa_\alpha \colon V_\alpha \to GL(n,k)\}_{\alpha}$ is defined, with refining functions $\phi \colon A \to I$ and $\phi' \colon A \to I'$. Let $\left\{ V_\alpha \times k^n \underoverset{\simeq}{\psi_{\phi(\alpha)}\vert_{V_\alpha} }{\to} E(c_1)\vert_{V_\alpha} \right\}_{\alpha \in A}$ and $\left\{ V_\alpha \times k^n \underoverset{\simeq}{\psi'_{\phi'(\alpha)}\vert_{V_\alpha} }{\to} E(c_2)\vert_{V_\alpha} \right\}_{\alpha \in A}$ be the corresponding restrictions of the canonical local trivilizations of the two glued bundles.

For $\alpha \in A$ define

$$f_\alpha \coloneqq \psi'_{\phi'(\alpha)}\vert_{V_\alpha} \circ \kappa_\alpha \circ (\psi_{\phi(\alpha)}\vert_{V_\alpha} )^{-1} \phantom{AAAA} \text{hence:} \phantom{AAA} \array{ V_\alpha \times k^n &\overset{ \psi_{\phi(\alpha)}\vert_{V_\alpha} }{\longrightarrow}& E(c_1)\vert_{V_\alpha} \\ {}^{\mathllap{\kappa_\alpha}}\downarrow && \downarrow^{\mathrlap{f_\alpha}} \\ V_\alpha \times k^n &\overset{ (\psi'_{\phi'(\alpha)}\vert_{V_\alpha})^{-1} }{\longleftarrow}& E(c_1)\vert_{V_\alpha} } \,.$$

Observe that for $\alpha, \beta \in A$ and $x \in V_\alpha \cap V_\beta$ the coboundary condition implies that

$$f_\alpha\vert_{V_\alpha \cap V_\beta} \;=\; f_\beta\vert_{V_\alpha \cap V_\beta}$$

because in the diagram

$$\array{ k^n &\overset{ g_{\phi(\alpha) \phi(\beta) }(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_\alpha(x)}}\downarrow && \downarrow^{\mathrlap{\kappa_{\beta}(x)}} \\ k^n &\underset{g'_{\phi'(\alpha) \phi'(\beta)}(x) }{\longrightarrow}& k^n } \phantom{AAAAA} = \phantom{AAAAA} \array{ k^n &\overset{ \psi_{\phi(\alpha)}(x) }{\longrightarrow}& E(c_1)_x &\overset{ (\psi_{\phi(\beta)})^{-1}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_\alpha(x)}}\downarrow && \downarrow^{\mathrlap{\exists !} } && \downarrow^{\mathrlap{\beta_\alpha(x)}} \\ k^n &\overset{ \psi'_{\phi'(\alpha)}(x) }{\longrightarrow}& E(c_2)_x &\overset{ (\psi'_{\phi'(\beta)})^{-1}(x) }{\longrightarrow}& k^n }$$

the vertical morphism in the middle on the right is unique, by the fact that all other morphisms in the diagram on the right are invertible.

Therefore there is a unique vector bundle homomorphism

$$f\;\colon\; E(c_1) \to E(c_2)$$

given for all $\alpha \in A$ by $f\vert_{V_\alpha} = f_\alpha$. Similarly there is a unique vector bundle homomorphism

$$f^{-1}\;\colon\; E(c_2) \to E(c_1)$$

given for all $\alpha \in A$ by $f^{-1}\vert_{V_\alpha} = f^{-1}_\alpha$. Hence this is the required vector bundle isomorphism.

Finally to see that the function from Cech cohomology classes to isomorphism classes of vector bundles thus defined is a bijection:

By prop. the function is surjective, and by prop. it is injective.

$\,$

Examples

Example

(tautological line bundle)

For $n \in \mathbb{N}$ then the projective space $k P^n$ carries the tautological line bundle whose fiber over the $k$-line $[v] \in k P^n$ is that $k$-line.

For details see there

Example

(cylinder)

Let

$$S^1 = \left\{ (x,y) \;\vert\; x^2 + y^2 = 1 \right\} \;\subset\, \mathbb{R}^2$$

be the circle with its Euclidean subspace metric topology.

Then the trivial real line bundle on the circle is the the cylinder

$$S^1 \times \mathbb{R}$$

Example

(Moebius strip)

Let

$$S^1 = \left\{ (x,y) \;\vert\; x^2 + y^2 = 1 \right\} \;\subset\, \mathbb{R}^2$$

be the circle with its Euclidean subspace metric topology. Consider the open cover

$$\left\{ U_n \subset S^1 \right\}_{n \in \{0,1,2\}}$$

with

$$U_n \coloneqq \left\{ (cos(\alpha), sin(\beta)) \;\vert\; n \frac{2 \pi }{3} - \epsilon \lt \alpha \lt (n+1) \frac{2\pi }{3} + \epsilon \right\}$$

for any $\epsilon \in (0,2\pi/6)$.

Define a Cech cohomology cocycle (remark ) on this cover by

$$g_{n_1 n_2} = \left\{ \array{ const_{-1} & \vert & (n_1,n_2) = (0,2) \\ const_{-1} &\vert& (n_1,n_2) = (2,0) \\ const_1 &\vert& \text{otherwise} } \right.$$

Since there are no non-trivial triple intersections, all cocycle conditions are evidently satisfied.

Accordingly by example these functions define a vector bundle. This is the Moebius strip

Example

(basic complex line bundle on the 2-sphere)

Let

$$S^2 \coloneqq \left\{ (x,y,z) \;\vert\; x^2 + y^2 + z^2 = 1 \right\} \subset \mathbb{R}^3$$

be the 2-sphere with its Euclidean subspace metric topology. Let

$$\left\{ U_{i} \subset S^2 \right\}_{i \in \{+,-\}}$$

be the two complements of antipodal points

$$U_\pm \coloneqq S^2 \setminus \{(0, 0, \pm 1)\} \,.$$

Define continuous functions

$$\array{ U_+ \cap U_- &\overset{g_{\pm \mp}}{\longrightarrow}& GL(1,\mathbb{C}) \\ ( \sqrt{1-z^2} \, cos(\alpha), \sqrt{1-z^2} \, sin(\alpha), z) &\mapsto& \exp(\pm 2\pi i \alpha) } \,.$$

Since there are no non-trivial triple intersections, the only cocycle condition is

$$g_{\mp \pm} g_{\pm \mp} = g_{\pm \pm} = id$$

which is clearly satisfied.

The complex line bundle this defined is called the basic complex line bundle on the 2-sphere.

With the 2-sphere identified with the complex projective space $\mathbb{C} P^1$ (the Riemann sphere), the basic complex line bundle is the tautological line bundle (example ) on $\mathbb{C}P^1$.

Example

(clutching construction)

Generally, for $n \in \mathbb{N}$, $n \geq 1$ then the n-sphere $S^n$ may be covered by two open hemispheres intersecting in an equator of the form $S^{n-1} \times (-\epsilon, \epsilon)$. A vector bundle is then defined by specifying a single function

$$g_{+-} \;\colon\; S^{n-1} \longrightarrow GL(n,k) \,.$$

This is called the clutching construction of vector bundles over n-spheres.

Example

(tangent bundle)

For $X$ the topological space underlyithe ng a differentiable manifold then its tangent bundle $T X$ is a real vector bundle over $X$ whose rank is the dimension of $X$.

Example

(normal bundle)

For $i X \hookrightarrow Y$ an embedding of differentiable manifolds, then the normal bundle

$$N_i X \coloneqq T Y/T X$$

is the real vector bundle over $Y$ whose fiber at $x \in X$ is the quotient vector space $(N_i X)_x \coloneqq T_{i(x)} Y / T_x X$.

Properties

Basic properties

Lemma

(homomorphism of vector bundles is isomorphism as soon as it is a fiberwise isomorphism)

Let $[E_1 \to X]$ and $[E_2 \to X]$ be two topological vector bundles (def. ).

If a homomorphism of vector bundles $f \colon E_1 \longrightarrow E_2$ restricts on the fiber over each point to a linear isomorphism

$$f\vert_x \;\colon\; (E_1)_x \overset{\simeq}{\longrightarrow} (E_2)_x$$

then $f$ is already an isomorphism of vector bundles.

Proof

It is clear that $f$ has an inverse function of underlying sets $f^{-1} \colon E_2 \to _E_1$ which is a function over $X$: Over each $x \in X$ it it the linear inverse $(f\vert_x)^{-1} \colon (E_2)_x \to (E_1)_x$.

What we need to show is that this is a continuous function.

By remark we find an open cover $\{U_i \subset X\}_{i \in I}$ over which both bundles have a local trivialization.

$$\left\{ U_i \underoverset{\simeq}{\phi^1_i}{\to} (E_1)\vert_{U_i}\right\}_{i \in I} \phantom{AA} \text{and} \phantom{AA} \left\{ U_i \underoverset{\simeq}{\phi^2_i}{\to} (E_2)\vert_{U_i} \right\}_{i \in I} \,.$$

Restricted to any patch $i \in I$ of this cover, the homomorphism $f|_{U_i}$ induces a homomorphism of trivial vector bundles

$$f_i \coloneqq \phi^2_j^{-1} \circ f \circ \phi^1_i \phantom{AAAAAA} \array{ U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& (E_1)\vert|_{U_i} \\ {}^{f_i}\downarrow && \downarrow^{\mathrlap{f\vert_{U_i}}} \\ U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& (E_2)\vert_{U_j} } \,.$$

Also the $f_i$ are fiberwise invertible, hence are continuous bijections. We claim that these are homeomorphisms, hence that their inverse functions $(f_i)^{-1}$ are also continuous.

To this end we re-write the $f_i$ a little. First observe that by the universal property of the product topological space and since they fix the base space $U_i$, the $f_i$ are equivalently given by a continuous function

$$h_i \;\colon\; U_i \times k^n \longrightarrow k^n$$

as

$$f_i(x,v) = (x, h_i(x,v)) \,.$$

Moreovern since $k^n$ is locally compact (like every finite dimensional vector space, by the Heine-Borel theorem), the mapping space adjunction says (by this prop.) that there is a continuous function

$$\tilde h_i \;\colon\; U_i \longrightarrow Maps(k^n, k^n)$$

(with $Maps(k^n,k^n)$ the set of continuous functions $k^n \to k^n$ equipped with the compact-open topology) which factors $h_i$ via the evaluation map as

$$h_i \;\colon\; U_i \times k^n \overset{\tilde h_i \times id_{k^n}}{\longrightarrow} Maps(k^n, k^n) \times k^n \overset{ev}{\longrightarrow} k^n \,.$$

By assumption of fiberwise linearity the functions $\tilde h_i$ in fact take values in the general linear group

$$GL(n,k) \subset Maps(k^n, k^n)$$

and this inclusion is a homeomorphism onto its image (by this prop.).

Since passing to inverse matrices

$$(-)^{-1} \;\colon\; GL(n,k) \longrightarrow GL(n,k)$$

is a rational function on its domain $GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}$ inside Euclidean space and since rational functions are continuous on their domain of definition, it follows that the inverse of $f_i$

$$(f_i)^{-1} \;\colon\; U_i \times k^n \overset{(id , \tilde h_i ) }{\longrightarrow} U_i \times k^n \times GL(n,k) \overset{ id \times (-)^{-1} }{\longrightarrow} U_i \times k^n \times GL(n,k) \overset{id \times ev}{\longrightarrow} U_i \times k^n$$

is a continuous function.

To conclude that also $f^{-1}$ is a continuous function we make use prop. to find an isomorphism between $E_2$ and a quotient topological space of the form

$$E_2 \simeq \left(\underset{i \in I}{\sqcup} (U_i \times k^n) \right) / \left( \left\{ g_{i j}\right\}_{i,j\in I} \right) \,.$$

Hence $f^{-1}$ is equivalently a function on this quotient space, and we need to show that as such it is continuous.

By the universal property of the disjoint union space (the coproduct in Top) the set of continuous functions

$$\{ U_i \times k^n \overset{f_i^{-1}}{\to} U_i \times k^n \overset{\phi^1_i}{\to} E_1 \}_{i \in I}$$

corresponds to a single continuous function of the form

$$(\phi^1_i \circ f_i^{-1})_{i \in I} \;\colon\; \underset{i \in I}{\sqcup} U_i \times k^n \longrightarrow E_1 \,.$$

These functions respect the equivalence relation, since for each $x \in U_i \cap U_j$ we have

$$(\phi^1_i \circ f_i^{-1})((x,i),v) = (\phi^1_j \circ f_j^{-1})( (x,j), g_{i j}(x)(v) ) \phantom{AAAA} \text{since:} \phantom{AAAA} \array{ && E_1 \\ & {}^{\mathllap{\phi^1_i \circ f_i^{-1}}}\nearrow & \uparrow^{\mathrlap{f^{-1}}} & \nwarrow^{\mathrlap{ \phi^1_j \circ f_j^{-1} }} \\ U_i \times k^n &\underset{\phi^2_i}{\longrightarrow}& (E_2)\vert_{U_i \cap U_i} &\underset{(\phi^2_j)^{-1}}{\longrightarrow}& U_i \times k^n } \,.$$

Therefore by the universal property of the quotient topological space $E_2$, these functions extend to a unique continuous function $E_2 \to E_1$ such that the following diagram commutes:

$$\array{ \underset{i \in i}{\sqcup} U_i \times k^n &\overset{( \phi^1_i \circ f_i^{-1} )_{i \in I}}{\longrightarrow}& E_1 \\ \downarrow & \nearrow_{\mathrlap{\exists !}} \\ E_2 } \,.$$

This unique function is clearly $f^{-1}$ (by pointwise inspection) and therefore $f^{-1}$ is continuous.

Example

(fiberwise linearly independent sections trivialize a vector bundle)

If a topological vector bundle $E \to X$ of rank $n$ admits $n$ sections (example )

$$\{\sigma_k \;\colon\; X \longrightarrow E\}_{k \in \{1, \cdots, n\}}$$

that are linearly independent at each point $x \in X$, then $E$ is trivializable (example ). In fact, with the sections regarded as vector bundle homomorphisms out of the trivial vector bundle of rank $n$ (according to example ), these sections are the trivialization

$$(\sigma_1, \cdots, \sigma_n) \;\colon\; (X \times k^n) \overset{\simeq}{\longrightarrow} E \,.$$

This is because their linear independence at each point means precisely that this morphism of vector bundles is a fiber-wise linear isomorphsm and therefore an isomorphism of vector bundles by lemma .

Direct summand bundles

We discuss properties of the direct sum of vector bundles for topological 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 (def. ).

Then every topological vector sub-bundle $E_1 \hookrightarrow E$ (example ) is a direct vector bundle summand, in that there exists another vector sub-bundle $E_2 \hookrightarrow E$ (example ) such that their direct sum of vector bundles 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 an inner product on vector bundles

$$\langle -,-\rangle \;\colon\; E \oplus_X E \longrightarrow X \times k$$

(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

(vector bundles over a compact Hausdorff space are direct summands of a trivial vector bundle)

Let

1. $X$ be a compact Hausdorff space;

2. $E \to X$ a topological vector bundle (def. ).

Then there exists another topological vector bundle $\tilde E \to X$ such that the direct sum of vector bundles of the two is isomorphic to a trivial vector bundle $X \times k^n$:

$$E \oplus \tilde E \;\simeq\; X \times k^n \,.$$

Proof

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

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

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

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

extend by 0 to vector bundle homomorphism of the form

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

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

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

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 in the analysis of topological K-theory groups on compact Hausdorff spaces. See there for more.

Concordance

We discuss that every concordance of topological vector bundles over a paracompact topological space makes the restriction of the vector bundle over the endpoints of the interval isomorphic (prop. below). In particular this implies tht the pullbacks of vector bundles along two homotopic continuous functions are isomorphic (corollary below).

This result is apparently due to Steenrod, see Theorem 11.4 in Steenrod. The proof below follows Hatcher, theorem 1.6.

For $X$ a topological space write $X \times I$ for the product topological space with the closed interval $[0,1]$ equipped with its Euclidean metric topology.

Write

$$X \overset{p_X}{\longleftarrow} X \times [0,1] \overset{p_{[0,1]}}{\longrightarrow} [0,1]$$

for the two continuous projections out of the product space.

Lemma

For $X$ a topological space, then a vector bundle $E \to X \times [0,1]$ is trivializable (example ) if its restrictions to $X \times [0,1/2]$ and to $X \times [1/2,1]$ are trivializable.

Lemma

For $X$ a topological space, then for every topological vector bundle $E \to X \times I$ there exists an open cover $\{U_i \subset X\}_{i \in I}$ of $X$ such that the vector bundle trivializes over $U_i \times [0,1] \subset X \times [0,1]$, for each $i \in I$.

Proof

By local trvializability of the vector bundle, there exists an open cover $\{V_j \subset X \times I\}_{j \in J}$ over which the bundle trivializes. For each point $x \in X$ this induces a cover of $\{x\} \times [0,1]$. This is a compact topological space (for instance by the Heine-Borel theorem) and hence there exists a finite subset $J_x \subset I$ such that $\{V_i \subset X \times I\}_{i \in J_x}$ still covers $\{x\} \times [0,1]$.

By finiteness of $J_x$, the intersection

$$U_x \coloneqq \underset{i \in J_x}{\cap} p_X(V_i)$$

is an open neighbourhood of $x$ in $X$. Moreover

$$\{ p_{[0,1]}(V_i) \subset I \}_{i \in J_x}$$

is an open cover of $[0,1]$ such that the given vector bundle trivializes over each element of $\{U_x \times p_{[0,1]}(V_i)\}_{i \in J_x}$.

By the nature of the Euclidean metric topology each open subset of $[0,1]$ is a union of intervals. So we may pass to a refinement of this cover of $[0,1]$ such that each element is a single interval. Again by compactness of $[0,1]$, this refinement has a finite subcover

$$\{W_{x,k} \subset [0,1]\}_{k \in K_x}$$

each element of which is an interval. Since this is a finite cover, we may find numbers $\{0 = t_0 \lt t_1 \lt t_2 \lt \cdots \lt t_{n_x} = 1\}$ such that

$$\{ [t_k, t_{k+1}] \subset [0,1] \}_{0 \leq k \lt n_x}$$

is a cover of $[0,1]$, and such that the given vector bundle still trivializes over $V_x \times [t_k, t_{k+1}]$ for all $0 \leq k \lt n_x$.

By lemma this implies that the vector bundle in fact trivializes over $U_x \times [0,1]$.

Applying this procedure for all points $x \in X$ yields a cover

$$\{ U_x \subset X \}_{x \in X}$$

with the required property.

Proposition

(concordance of topological vector bundles)

Let $X$ be a paracompact Hausdorff space. If $E \to X \times [0,1]$ is a topological vector bundle over the product space of $X$ with the closed interval (hence a concordance of topological vector bundles on $X$), then the two endpoint-restrictions

$$E|_{X \times \{0\}} \phantom{AA} \text{and} \phantom{AA} E|_{X \times \{1\}}$$

are isomorphic vector bundles over $X$.

Proof

By lemma there exists an open cover $\{U_i \subset X\}_{i \in I}$ of $X$ such that the vector bundle $E$ trivializes over $U_i \times [0,1]$ for each $i \in I$. By this lemma there exists a countable cover

$$\{V_n \subset X\}_{n \in \mathbb{N}}$$

such that each element is a disjoint union of open subsets that each are contained in one of the $U_i$. This means that the vector bundle $E$ still trivializes over $V_n \times [0,1]$, for each $n \in \mathbb{N}$.

Moreover, since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity, there exists a partition of unity $\left\{f_n \colon X \to [0,1] \right\}_{n \in \mathbb{N}}$ subordinate to this countable cover.

For $n \in \mathbb{N}$ define

$$\psi_n \coloneqq \underoverset{k = 0}{n}{\sum} f_n$$

(so $\psi_0 = 0$ and by local finiteness there is for each $x \in X$ an $n_x$ such that $\psi_{n \gt n_x} = 1$.)

Now write

$$X_n \coloneqq graph( \psi_n ) \subset X \times [0,1]$$

for the graph of the function $\psi_n$ equipped with its subspace topology, and write

$$E_n \coloneqq \psi_n^\ast E$$

for the restriction of $E$ to that subspace

$$\array{ E_n &\longrightarrow& E \\ \downarrow && \downarrow \\ X_n = graph(\psi_n) &\hookrightarrow& X }$$

Observe that the projection functions

$$\array{ p_{n+1,n} \colon & X_{n+1} &\overset{}{\longrightarrow}& X_n \\ & (x,\psi_{n+1}(x)) &\overset{\phantom{AA}}{\mapsto}& (x, \psi_n(x)) = (x, \psi_{n+1}(x) - f_{n+1}(x)) }$$

are continuous functions: By the nature of the product topology and the subspace topology it is sufficient to check for $U \subset X$ and $V \subset \mathbb{R}$ open subsets, that every point $(x,c)$ in the preimage $p_n^{-1}( U \times V ) \subset X \times [0,1]$ is contained in an open subset of the form $U_x \times V_x \subset X \times [0,1]$ such that every point of $X_{n+1}$ that is also in $U_x \times V_x$ is still mapped to $U \times V$. Such an open subset is $\left( U \cap \psi_n^{-1}(V) \right) \times [0,1]$.

Also observe that the composites

$$E_n \longrightarrow X_n \overset{p_{n,0}}{\longrightarrow} X_0 = 0$$

make each $E_n$ a vector bundle over $X$: To see local trivializability over $X$ choose a local trivialization of $E$ over some open cover $\{U_i \subset X\}_{i \in I}$ and observe that then $E_n$ is trivial over the fiber product$X_n \times_X U_n$ and hence over $U_n$.

Now by the pullback definition of the $E_n$, the pasting law says that for each $n \in \mathbb{N}$ we have a pullback square of vector bundles of the form

$$\array{ E_{n+1} && \overset{h_n}{\longrightarrow} && E_n \\ \downarrow && (pb) && \downarrow \\ X_{n+1} && \longrightarrow && X_n \\ & \searrow && \swarrow \\ && X } \,.$$

By the nature of pullbacks, the top horizontal function $h_n$ in this diagram is on each fiber a linear isomorphism. Therefore prop. implies that each $h_n$ is in fact an isomorphism of vector bundles over $X$

By local finiteness, each point $x \in X$ has a neighbourhood $U_x$ such that only a finite number $n_x$ of these $h_n$ are non-trivial, and so it makes sense to consider the infinite composition

$$h \coloneqq h_1 \circ h_2 \circ h_3 \circ \cdots$$

understood to be on each $U_x$ the finite composite

$$h(x) \coloneqq h_1 \circ \cdots \circ h_{n_x} \,.$$

Since all the $h_k$ are vector bundle isomorphisms, so are all their composites. Thus $h$ is an isomorphism of the required form

$$h \;\colon\; E|_{X \times \{0\}} \overset{\simeq}{\longrightarrow} E|_{X \times \{1\}} \,.$$

Corollary

Let $X$ be a paracompact Hausdorff space, let $E \to Y$ be a topological vector bundle, let $f,g \colon X \to Y$ be two continuous functions, and let $\eta \colon f \to g$ be a left homotopy between them. Then there is an isomorphism of vector bundles over $X$ between the pullback of vector bundles of $E$ along $f$ and along $g$, respectively:

$$f^\ast E \simeq g^\ast E \,.$$

Proof

By definition, the left homotopy $\eta$ is a continuous function of the form

$$\eta \;\colon\; X \times [0,1] \longrightarrow Y \,.$$

For $t \in [0,1]$ write $i_t$ for the continuous function

$$\array{ X &\overset{\phantom{AA}i_t\phantom{AA}}{\longrightarrow}& X \times [0,1] \\ x &\overset{\phantom{AAAA}}{\mapsto}& (x,t) } \,.$$

By the pasting law for pullbacks we have that

$$f^\ast E = (\eta \circ i_0)^\ast E \simeq i_0^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{0\}}$$

and

$$g^\ast E = (\eta \circ i_1)^\ast E \simeq i_1^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{1\}}$$

With this the statement follows by prop. .

Example

(homotopy invariance of isomorphism classes of vector bundles)

Let $X$ and $Y$ be paracompact Hausdorff spaces and let

$$f \;\colon\; X \longrightarrow Y$$

be a continuous function which is a homotopy equivalence. Then pullback along $f$ constitutes a bijection on sets of isomorphism classes of topological vector bundles:

$$f^\ast \;\colon\; Vect(Y)/_\sim \overset{\simeq}{\longrightarrow} Vect(X)/_\sim \,.$$

Proof

By definition of homotopy equivalence, there is a continuous function $g \colon Y \longrightarrow X$ and left homotopies

$$g \circ f \Rightarrow id \phantom{AAAA} f \circ g \Rightarrow id \,.$$

Hence corollary implies that

$$f^\ast \circ g^\ast = (g \circ f)^\ast = id \phantom{AAAAA} g^\ast \circ f^\ast = (f \circ g)^\ast = id \,.$$

This mean that $g^\ast$ is the inverse function to $f^\ast$, and hence both are bijections.

Example

(topological vector bundle on contractible topological space is trivializable)

If $X$ is a contractible topological space, then every topological vector bundle over $X$ is isomorphic to a trivial vector bundle.

Proof

That $X$ is contractible means by definition that there is a left homotopy of the form

$$\array{ X &\longrightarrow& \ast \\ \mathllap{i_0}\downarrow & & \downarrow \\ X \times [0,1] &\overset{\eta}{\longrightarrow}& X \\ \mathllap{i_1}\uparrow & \nearrow_{\mathrlap{id}} \\ X & } \,.$$

By cor it follows that for $E \to X$ any topological vector bundle that there is an isomorphism between $id^\ast E = E$ and the result of first restricting the bundle to the point, and then forming the pullback bundle along $X \to \ast$. But the latter operation precisely produces the trivial vector bundles over $X$.

Over closed subspaces

We discuss the behavour of vector bundles with respect to closed subspaces $A \subset X$ of compact Hausdorff spaces.

Lemma

(isomorphism of vector bundles on closed subset of compact Hausdorff spaces extends to open neighbourhood)

Let $k \in \{\mathbb{R}, \mathbb{C}\}$, let $X$ be a compact Hausdorff space and let $A \subset X$ a closed subspace. Let $E_i \overset{p_i}{\to} X$ be two topological vector bundles over $X$, $i \in \{1,2\}$.

If there exists an isomorphism

$$E_1\vert_A \overset{\simeq}{\longrightarrow} E_2\vert_A$$

of the restricted vector bundles over $A$, then there also exists an open subset $U \subset X$ with $A \subset U$ such that there is also an isomorphism

$$E_1\vert_U \overset{\simeq}{\longrightarrow} E_2\vert_U$$

of the vector bundles restricted to $U$.

Proof

A bundle isomorphism $E_1\vert_A \simeq E_2\vert_A$ is equivalently a trivializing section (example ) of the tensor product of vector bundles $(E_1\vert_A)^\ast \otimes_A E_2\vert_A$ of $E_2\vert_A$ with the dual vector bundle $(E_2\vert_A)^\ast$. (by this prop.).

Let $\{V_i \subset X\}_{i \in I}$ be an open cover of $X$ over which this tensor product bundle trivializes with trivializations

$$\left\{ V_i \times \mathbb{R}^{(n^2)} \underoverset{\simeq}{\phi_i}{\longrightarrow} (E_1^\ast \otimes_X E_2)\vert_{U_i} \right\} \,.$$

Since compact Hausdorff spaces are normal, the shrinking lemma applies and gives a refinement of this by a cover $\{U_i \subset X\}_{i \in I}$ by closed subsets $U_i \subset X$.

Then a trivializing section $\sigma \in \Gamma_A\left( (E_1\vert_A)^\ast \otimes_A E_2 \vert_A \right)$ as above is on each $U_i \cap A$ a continuous function

$$\sigma_i \;\colon\; U_i \cap A \longrightarrow GL(n,k)$$

to the general linear group $GL(n,k) \subset Mat_{n \times n}(k)$, such that

$$\sigma\vert_{U_i \cap A} = \phi_i \circ \sigma_i \,.$$

Regarded as a function to the $n \times n$ matrices, this is a set of $n^2$ continuous function $((\sigma_i)_{a b})$

Now since $U_i \subset X$ is closed by construction, and $A \subset X$ is closed by assumption, also the intersections $U_i \cap X$ are closed. Since compact Hausdorff spaces are normal the Tietze extension theorem therefore applies to these component functions and yields extensions of each $\sigma_i$ to a continuous function of the form

$$\hat \sigma_i \;\colon\; U_i \longrightarrow Mat_{n \times n}(k) \,.$$

Moreover, since compact Hausdorff spaces are evidently paracompact Hausdorff spaces, and since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity, it follows that we find a partition of unity $\{f_i \colon U_i \to \mathbb{R} \}_{i \in I}$.

Consider then the functions $f_i \cdot \hat \sigma_i$ given by pointwise multiplication and regarded, via extension by zero, as continuous functions on all of $X$

$$f_i \cdot \hat \sigma_i \;\colon\; X \longrightarrow \mathbb{R} \,.$$

Summing these up yields a single section $\hat \sigma$ of $E_1^\ast \otimes_X E_2$

$$\hat \sigma \coloneqq \sum_{i \in I} \phi_i(f_i \cdot \hat \sigma_i) \in \Gamma_X(E_1^\ast \otimes_X E_2) \,,$$

which by construction is an extension of the original section, in that

$$\hat \sigma\vert_A = \sigma \,.$$

This is because for each $a \in A \subset X$ we have, using the above definitions,

$$\begin{aligned} \left(\underset{i \in I}{\sum} \phi_i(f_i \cdot \hat \sigma_i)\right)(a) & = \underset{i \in I}{\sum} (\phi_i (\hat \sigma_i(a))) \\ & = \underset{i \in I}{\sum} \phi_i( f_i(a) \sigma_i(a) ) \\ & = \underset{i \in I}{\sum} f_i(a) \cdot (\phi_i \circ \sigma_i)(a) \\ & = \underset{i \in I}{\sum} f_i(a) \cdot \sigma(a) \\ & = \left( \underset{i \in I}{\sum} f_i(a)\right) \cdot \sigma(a) \\ & = \sigma(a) \end{aligned}$$

Here the last step uses the nature of the partition of unity.

Now while $\hat \sigma$ is an extension of the section $\sigma$ to $X$, it will in general not be a trivializing section on $X$.

But since the general linear group $GL(n,k) = det^{-1}(k \setminus \{0\}) \subset Mat_{n \times n}(k)$ is an open subset of the Euclidean space $Mat_{n \times n}(k) \simeq k^{(n^2)}$, it follows that each point $x \in A$ has an open neighbourhood $U_x \subset X$ such that $\hat \sigma\vert_{U_x}$ is still a trivializing section, namely choosing $i_x \in I$ such that $x \in U_{i_x}$ set

$$U_x \coloneqq (\hat \sigma_{i_x})^{-1}( GL(n,k) ) \,.$$

The union of these

$$U \coloneqq \underset{x \in A}{\cup} U_x$$

is hence an open subset containing $A$ such that $(E_1^\ast \otimes_X E_2)\vert_U$ has a trivializing section, extending $\sigma$, hence such that there is an isomorphism $E_1\vert_U \simeq E_2 \vert_U$ extending the original isomorphism on $A$.

As a consequence:

Proposition

(vector bundle trivial over closed subspace of compact Hausdorff space is pullback of bundle on quotient space)

Let $X$ be a compact Hausdorff space and let $A \subset X$ be a closed subspace.

If a topological vector bundle $E \overset{p}{\to} X$ is such that its restriction $E\vert_A$ is trivializable, then $E$ is isomorphic to the pullback bundle $q^\ast E'$ of a topological vector bundle $E' \to X/A$ over the quotient space.

Proof

Let

$$A \times k^n \underoverset{\simeq}{\phi}{\longrightarrow} E\vert_A$$

be an isomorphism of vector bundles over $A$, which exists by assumption. Consider then on the total space $E\vert_A$ the equivalence relation given by

$$\phi^{-1}(x,v) \sim \phi^{-1}(x',v)$$

for all $x,x' \in A$ and $v \in k^n$. Let

$$E' \coloneqq E/\sim$$

be the corresponding quotient topological space. Observe that for $x \in X$ we have $E'_x = E_x$ while for $x \in A$ we have a canonical identification $E'_{x/A} \simeq k^n$, and over these points quotient coprojection is identified with $\phi^{-1}$:

$$\array{ E &\overset{}{\longrightarrow}& E' \\ (x,v) &\mapsto& \left\{ \array{ (x,v) &\vert& x \in X \setminus A \\ \phi^{-1}_x(v) &\vert& x\in A } \right. } \,.$$

Since the composite continuous function

$$E \overset{p}{\longrightarrow} X \overset{q}{\longrightarrow} X/A$$

respects the equivalence relation (in that it sends any two equivalent points to the same image point) the universal property of the quotient space yields a continuous function

$$p' \;\colon\; E' \to X/A$$

such that the following diagram commutes

$$\array{ E &\longrightarrow& E' \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{p'}} \\ X &\overset{q}{\longrightarrow}& X/A } \,.$$

We claim that this is a pullback diagram in Top:

By the above description of the top horizontal function, it is a pullback diagram of underlying sets. Hence we need to see that the topology on $E$ has a base given by the pre-images of the open subsets in $X$ and in $E'$. Now by definition of the quotient space topology on $E'$, its open subsets are those of $E$ that either do not contain a point $(x,v)$ with $x \in A$ or if they do, then they also contain all the points of the form $(x', \phi_{x'}^{-1}(\phi_x(v)))$ for $x' \in A$. Moreover, if $(x,v)$ is in the open subset for $x \in A$, then also $(x,v')$ for all $v'$ in some open ball in $k^n$ containing $v$. Hence intersecing these pre-images with pre-images of open subsets of $X$ under $p$ yields a basis for the topology.

Hence it only remains to see that $E' \overset{p'}{\longrightarrow} X/A$ is a vector bundle. The fiberwise linearity is clear, we need to show that it is locally trivializable.

To that end, let $\{U_i \subset X\}_{i \in I}$ be an open cover over which $E \overset{p}{\to} X$ has a local trivialization. Since $A \subset X$ is assumed to be closed, it follows that

$$\left\{ U_i \setminus A \subset X \setminus A\right\}_{i \in I}$$

is an open cover of the complement of $A$ in $X$. By the nature of the quotient space topology, this induces an open cover of $X\setminus A$. If we adjoin the quotient $U/A$ of an open neighbourhood $U \subset X$ of $A$ in $X$, then

$$\{ U_i \setminus A \subset X/A \} \sqcup \{ U/A \subset X/A \}$$

is an open cover of $X/A$. Moreover, by the construction of $E' \overset{p'}{\to} X/A$ it is clear that this bundle has a local trivialization over $U_i$, since $E \overset{p}{\to} X$ does, and similarly $E'$ trivializes over $U/A$ if $E$ trivializes over $U$. But such a $U$ does indeed exist by lemma .

Remark

Prop is the reason why reduced topological K-theory satisfies the long exact sequences in cohomology that make it a generalized (Eilenberg-Steenrod) cohomology theory. See

Proposition

Let $X$ be a compact Hausdorff space and $A \subset X$ a closed subspace and write $X/A$ for the corresponding quotient topological space (this example) with quotient coprojection denoted $q \colon X \longrightarrow X/A$.

If $A$ is a contractible topological space then the pullback bundle construction

$$q^\ast \;\colon\; Vect(X/A)_{/\sim} \longrightarrow Vect(X)_{/\sim}$$

is an isomorphism.

Proof

By example every vector bundle $E \overset{p}{\to} X$ is trivializable over the contractible subspace $A$. Therefore prop. implies that it is in the image of the pullback bundle map $q^\ast$. This says that $q^\ast$ is surjective. Finally, it is clear that it is injective. Therefore it is bijective.

Example

Let $(X,x)$ be a pointed compact topological space.

For $[0,1] \subset \mathbb{R}$ the closed interval with its Euclidean metric topology.

There is

1. the ordinary cylinder, being the product space $X \times I$

2. the reduced cylinder $X \wedge I_+ = (X \times I)/( \{x\} \times I )$ which is the smash product with the interval that has a base point freely adjoined

and

1. the ordinary suspension $S X \coloneqq (X \times I)/( X \times \{0,1\} )$;

2. the reduced suspension $\Sigma X \coloneqq (S X)/( \{x\} \times I )$.

In both cases the reduced space is obtained from the unreduced space by quotienting out the contractible closed subspace $I \simeq \{x\} \times I$ and hence topological vector bundles do not see the difference between the reduced and the unreduced spaces, by prop. .

Related concepts

• algebraic vector bundle

• differentiable vector bundle

• topological K-theory

References

The original reference for many results about bundles, including the theorem that concordance implies isomorphism, is

• Norman Steenrod, The Topology of Fibre Bundles, Princeton University Press (1951, 1957, 1960) [jstor:j.ctt1bpm9t5]

Further textbook accounts:

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) [ISBN:9780691081229, doi:10.1515/9781400881826, pdf]

• Glenys Luke, Alexander S. Mishchenko, Vector bundles and their applications, Math. and its Appl. 447, Kluwer 1998. viii+254 pp. MR99m:55019

• Dale Husemoeller, Michael Joachim, Branislav Jurco, Martin Schottenloher, Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics, Springer 2008 (pdf)

Lecture notes with an eye towards topological K-theory:

• Klaus Wirthmüller, Vector bundles and K-theory, 2012 (pdf)

• Allen Hatcher, chapter 1 of Vector bundles and K-Theory, (partly finished book) web