9.2 Fibre bundles

The tangent bundle in the previous section is an example of a more general framework called a fibre bundle. Definitions are now in order.

9.2.1 Definitions

Definition 9.1

A (differentiable) fibre bundle $(E,\pi ,M,F,G)$ consists of the following elements:

(i) A differentiable manifold $E$ called the total space.

(ii) A differentiable manifold $M$ called the base space.

(iii) A differentiable manifold $F$ called the fibre (or typical fibre).

(iv) A surjection $\pi :E\to M$ called the projection. The inverse image ${\pi }^{-1}(p)=F_p\cong F$ is called the fibre at $p$ .

(v) A Lie group $G$ called the structure group, which acts on $F$ on the left.

(vi) A set of open covering $\{U_i\}$ of $M$ with a diffeomorphism ${\varphi }_i:U_i\times F\to {\pi }^{-1}(U_i)$ such that $\pi \circ {\varphi }_i(p,f)=p$ . The map ${\varphi }_i$ is called the local trivialization since ${\varphi }_i^{-1}$ maps ${\pi }^{-1}(U_i)$ onto the direct product $U_i\times F$ .

(vii) If we write ${\varphi }_i(p,f)={\varphi }_{i,p}(f)$ , the map ${\varphi }_{i,p}:F\to F_p$ is a diffeomorphism. On $U_i\cap U_j\ne \varnothing$ , we require that $t_{ij}(p)\equiv {\varphi }_{i,p}^{-1}\circ {\varphi }_{j,p}:F\to F$ be an element of $G$ . Then ${\varphi }_i$ and ${\varphi }_j$ are related by a smooth map $t_{ij}:U_i\cap U_j\to G$ as (figure 9.2)

Figure 9.2. On the overlap $U_i\cap U_j$ , two elements $f_i,f_j\in F$ are assigned to $u\in {\pi }^{-1}(p)$ , $p\in U_i\cap U_j$ . They are related by $t_{ij}(p)$ as $f_i=t_{ij}(p)f_j$ .

$${\varphi }_j(p,f)={\varphi }_i(p,t_{ij}(p)f).\text{\hspace{1em}(9.4)}$$

The maps $t_{ij}$ are called the transition functions.

[Remarks: We often use a shorthand notation $E\stackrel{\pi }{\to }M$ or simply $E$ to denote a fibre bundle $(E,\pi ,M,F,G)$ .]

Strictly speaking, the definition of a fibre bundle should be independent of the special covering $\{U_i\}$ of $M$ . In the mathematical literature, this definition is employed to define a coordinate bundle $(E,\pi ,M,F,G,\{U_i\},\{{\varphi }_i\})$ . Two coordinate bundles $(E,\pi ,M,F,G,\{U_i\},\{{\varphi }_i\})$ and $(E,\pi ,M,F,G,\{V_i\},\{{\psi }_i\})$ are said to be equivalent if $(E,\pi ,M,F,G,\{U_i\}\cup \{V_j\},\{{\varphi }_i\}\cup \{{\psi }_j\})$ is again a coordinate bundle. A fibre bundle is defined as an equivalence class of coordinate bundles. In practical applications in physics, however, we always employ a certain

definite covering and make no distinction between a coordinate bundle and a fibre bundle.]

We need to clarify several points. Let us take a chart $U_i$ of the base space $M$ . ${\pi }^{-1}(U_i)$ is a direct product diffeomorphic to $U_i\times F$ , ${\varphi }_i^{-1}:{\pi }^{-1}(U_i)\to U_i\times F$ being the diffeomorphism. If $U_i\cap U_j\ne \varnothing$ , we have two maps ${\varphi }_i$ and ${\varphi }_j$ on $U_i\cap U_j$ . Let us take a point $u$ such that $\pi (u)=p\in U_i\cap U_j$ . We then assign two elements of $F$ , one by ${\varphi }_i^{-1}$ and the other by ${\varphi }_j^{-1}$ ,

$${\varphi }_i^{-1}(u)=\left(p,f_i\right),{\varphi }_j^{-1}(u)=\left(p,f_j\right)\text{\hspace{1em}(9.5)}$$

see figure 9.2. There exists a map $t_{ij}:U_i\cap U_j\to G$ which relates $f_i$ and $f_j$ as $f_i=t_{ij}(p)f_j$ . This is also written as (9.4).

We require that the transition functions satisfy the following consistency conditions:

$$t_{ii}(p)=\text{i d e n t i t y m a p}(p\in U_i)\text{\hspace{1em}(9.6a)}$$

$$t_{ij}(p)=t_{ji}(p{)}^{-1}(p\in U_i\cap U_j)\text{\hspace{1em}(9.6b)}$$

$$t_{ij}(p)\cdot t_{jk}(p)=t_{ik}(p)(p\in U_i\cap U_j\cap U_k).\text{\hspace{1em}(9.6c)}$$

Unless these conditions are satisfied, local pieces of a fibre bundle cannot be glued together consistently. If all the transition functions can be taken to be identity maps, the fibre bundle is called a trivial bundle. A trivial bundle is a direct product $M\times F$ .

Given a fibre bundle $E\stackrel{\pi }{\to }M$ , the possible set of transition functions is obviously far from unique. Let $\{U_i\}$ be a covering of $M$ and $\{{\varphi }_i\}$ and $\{{\tilde{\varphi }}_i\}$ be two sets of local trivializations giving rise to the same fibre bundle. The transition functions of respective local trivializations are

$$t_{ij}(p)={\varphi }_{i,p}^{-1}\circ {\varphi }_{j,p}\text{\hspace{1em}(9.7a)}$$

$${\tilde{t}}_{ij}(p)={\tilde{\varphi }}_{i,p}^{-1}\circ {\tilde{\varphi }}_{j,p}.\text{\hspace{1em}(9.7b)}$$

Define a map $g_i(p):F\to F$ at each point $p\in M$ by

$$g_i(p)\equiv {\varphi }_{i,p}^{-1}\circ {\tilde{\varphi }}_{i,p}.\text{\hspace{1em}(9.8)}$$

We require that $g_i(p)$ be a homeomorphism which belongs to $G$ . This requirement must certainly be fulfilled if $\{{\varphi }_i\}$ and $\{{\tilde{\varphi }}_i\}$ describe the same fibre bundle. It is easily seen from (9.7) and (9.8) that

$${\tilde{t}}_{ij}(p)=g_i(p{)}^{-1}\circ t_{ij}(p)\circ g_j(p).\text{\hspace{1em}(9.9)}$$

In the practical situations which we shall encounter later, $t_{ij}$ are the gauge transformations required for pasting local charts together, while $g_i$ corresponds to the gauge degrees of freedom within a chart $U_i$ . If the bundle is trivial, we may put all the transition functions to be identity maps. Then the most general form of the transition functions is

$$t_{ij}(p)=g_i(p{)}^{-1}{g}_j(p).\text{\hspace{1em}(9.10)}$$

Let $E\stackrel{\pi }{\to }M$ be a fibre bundle. A section (or a cross section) $s:M\to E$ is a smooth map which satisfies $\pi \circ s={\mathrm{id}}_M$ . Clearly, $s(p)=s{|}_p$ is an element of $F_p={\pi }^{-1}(p)$ . The set of sections on $M$ is denoted by $\Gamma (M,F)$ . If $U\subset M$ , we may talk of a local section which is defined only on $U$ . $\Gamma (U,F)$ denotes the set of local sections on $U$ . For example, $\Gamma (M,TM)$ is identified with the set of vector fields $\mathcal{X}(M)$ . It should be noted that not all fibre bundles admit global sections.

Example 9.1

Let $E$ be a fibre bundle $E\stackrel{\pi }{\to }{S}^1$ with a typical fibre $F=[-1,1]$ . Let $U_1=(0,2\pi )$ and $U_2=(-\pi ,\pi )$ be an open covering of $S^1$ and let $A=(0,\pi )$ and $B=(\pi ,2\pi )$ be the intersection $U_1\cap U_2$ , see figure 9.3. The local trivializations ${\varphi }_1$ and ${\varphi }_2$ are given by

Figure 9.3. The base space $S^1$ and two charts $U_1$ and $U_2$ over which the fibre bundle is trivial.

$${\varphi }_1^{-1}(u)=(\theta ,t),{\varphi }_2^{-1}(u)=(\theta ,t)$$

for $\theta \in A$ and $t\in F$ . The transition function $t_{12}(\theta )$ , $\theta \in A$ , is the identity map $t_{12}(\theta ):t\mapsto t$ . We have two choices on $B$ ;

(I) ${\varphi }_1^{-1}(u)=(\theta ,t),{\varphi }_2^{-1}(u)=(\theta ,t)$ (II) ${\varphi }_1^{-1}(u)=(\theta ,t),{\varphi }_2^{-1}(u)=(\theta ,-t)$

For case (I), we find that $t_{12}(\theta )$ is the identity map and two pieces of the local bundles are glued together to form a cylinder (figure 9.4(a)). For case (II), we have $t_{12}(\theta ):t\mapsto -t,\theta \in B$ , and obtain the Möbius strip (figure 9.4(b)). Thus, a cylinder has the trivial structure group $G=\{e\}$ where $e$ is the identity map of $F$ onto $F$ while the Möbius strip has $G=\{e,g\}$ where $g:t\mapsto -t$ . Since $g^2=e$ , we find $G\cong {\mathbb{Z}}_2$ . A cylinder is a trivial bundle $S^1\times F$ , while the Möbius strip is not. [Remark: The group ${\mathbb{Z}}_2$ is not a Lie group. This is the only occasion we use a discrete group for the structure group.]

(a)

(b)

Figure 9.4. Two fibre bundles over $S^1$ : (a) is the cylinder which is a trivial bundle $S^1\times I$ ; (b) is the Möbius strip.

9.2.2 Reconstruction of fibre bundles

What is the minimal information required to construct a fibre bundle? We now show that for given $M$ , $\{U_i\}$ , $t_{ij}(p)$ , $F$ and $G$ , we can reconstruct the fibre bundle $(E,\pi ,M,F,G)$ . This amounts to finding a unique $\pi$ , $E$ and ${\varphi }_i$ from given data. Let us define

$$X\equiv \bigcup _i{U}_i\times F.\text{\hspace{1em}(9.11)}$$

Introduce an equivalence relation $\sim$ between $(p,f)\in U_i\times F$ and $(q,f^{\prime })\in U_j\times F$ by $(p,f)\sim (q,f^{\prime })$ if and only if $p=q$ and $f^{\prime }=t_{ij}(p)f$ . A fibre

bundle $E$ is then defined as

$$E=X/\sim .\text{\hspace{1em}(9.12)}$$

Denote an element of $E$ by $[(p,f)]$ . The projection is given by

$$\pi :[(p,f)]\mapsto p.\text{\hspace{1em}(9.13)}$$

The local trivialization ${\varphi }_i:U_i\times F\to {\pi }^{-1}(U_i)$ is given by

$${\varphi }_i:(p,f)\mapsto [(p,f)].\text{\hspace{1em}(9.14)}$$

The reader should verify that $E,\pi$ and $\{{\varphi }_i\}$ thus defined satisfy all the axioms of fibre bundles. Thus, the given data reconstruct a fibre bundle $E$ uniquely.

This procedure may be employed to construct a new fibre bundle from an old one. Let $(E,\pi ,M,F,G)$ be a fibre bundle. Associated with this bundle is a new bundle whose base space is $M$ , transition function $t_{ij}(p)$ , structure group $G$ and fibre $F^{\prime }$ on which $G$ acts. Examples of associated bundles will be given later.

9.2.3 Bundle maps

Let $E\stackrel{\pi }{\to }M$ and $E^{\prime }\stackrel{{\pi }^{\prime }}{\to }{M}^{\prime }$ be fibre bundles. A smooth map $\bar{f}:E^{\prime }\to E$ is called a bundle map if it maps each fibre $F_p^{\prime }$ of $E^{\prime }$ onto $F_q$ of $E$ . Then $\bar{f}$ naturally induces a smooth map $f:M^{\prime }\to M$ such that $f(p)=q$ (figure 9.5). Observe that the diagram

Figure 9.5. A bundle map $\bar{f}:E^{\prime }\to E$ induces a map $f:M^{\prime }\to M$ .

$$\begin{array}{ccc}{E}^{\prime }& \stackrel{\bar{f}}{⟶}& E\\ {\pi }^{\prime }\downarrow & & \downarrow \pi \\ M^{\prime }& \stackrel{f}{⟶}& M\end{array}\left(\begin{array}{ccc}u& \stackrel{\bar{f}}{⟶}& \bar{f}(u)\\ {\pi }^{\prime }\downarrow & & \downarrow \pi \\ p& \stackrel{f}{⟶}& q\end{array}\right)\text{\hspace{1em}(9.15)}$$

commutes. [Caution: A smooth map $\bar{f}:E^{\prime }\to E$ is not necessarily a bundle map. It may map $u,v\in F_p^{\prime }$ of $E^{\prime }$ to $\bar{f}(u)$ and $\bar{f}(v)$ on different fibres of $E$ so that $\pi (\bar{f}(u))\ne \pi (\bar{f}(v))$ .]

9.2.4 Equivalent bundles

Two bundles $E^{\prime }\stackrel{{\pi }^{\prime }}{\to }M$ and $E\stackrel{\pi }{\to }M$ are equivalent if there exists a bundle map $\bar{f}:E^{\prime }\to E$ such that $f:M\to M$ is the identity map and $\bar{f}$ is a diffeomorphism:

$$\begin{array}{ccc}{E}^{\prime }& \stackrel{\bar{f}}{⟶}& E\\ {\pi }^{\prime }\downarrow & & \downarrow \pi \\ M& \stackrel{{\mathrm{id}}_M}{⟶}& M.\end{array}\text{\hspace{1em}(9.16)}$$

This definition of equivalent bundles is in harmony with that given in the remarks following definition 9.1.

9.2.5 Pullback bundles

Let $E\stackrel{\pi }{\to }M$ be a fibre bundle with typical fibre $F$ . If a map $f:N\to M$ is given, the pair $(E,f)$ defines a new fibre bundle over $N$ with the same fibre $F$ (figure 9.6). Let $f^{\ast }E$ be a subspace of $N\times E$ , which consists of points $(p,u)$ such that $f(p)=\pi (u)$ . $f^{\ast }E\equiv \{(p,u)\in N\times E|f(p)=\pi (u)\}$ is called the pullback of $E$ by $f$ . The fibre $F_p$ of $f^{\ast }E$ is just a copy of the fibre $F_{f(p)}$ of $E$ . If we define $f^{\ast }E\stackrel{{\pi }_1}{\to }N$ by ${\pi }_1:(p,u)\mapsto p$ and $f^{\ast }E\stackrel{{\pi }_2}{\to }E$ by $(p,u)\mapsto u$ , the pullback $f^{\ast }E$ may be endowed with the structure of a fibre bundle and we obtain the following bundle map,

Figure 9.6. Given a fibre bundle $E\stackrel{\pi }{\to }M$ , a map $f:N\to M$ defines a pullback bundle $f^{\ast }E$ over $N$ .

$$\begin{array}{ccc}{f}^{\ast }E& \stackrel{{\pi }_2}{\to }& E\\ {\pi }_1\downarrow & & \downarrow \pi \\ N& \stackrel{f}{\to }& M\end{array}\left(\begin{array}{ccc}(p,u)& \stackrel{{\pi }_2}{\to }& u\\ {\pi }_1\downarrow & & \downarrow \pi \\ p& \stackrel{f}{\to }& f(p)\end{array}\right).\text{\hspace{1em}(9.17)}$$

The commutativity of the diagram follows since $\pi ({\pi }_2(p,u))=\pi (u)=f(p)=f({\pi }_1(p,u))$ for $(p,u)\in f^{\ast }E$ . In particular, if $N=M$ and $f={\mathrm{id}}_M$ , then two fibre bundles $f^{\ast }E$ and $E$ are equivalent.

Let $\{U_i\}$ be a covering of $M$ and $\{{\varphi }_i\}$ be local trivializations. $\{f^{-1}(U_i)\}$ defines a covering of $N$ such that $f^{\ast }E$ is locally trivial. Take $u\in E$ such that $\pi (u)=f(p)\in U_i$ for some $p\in N$ . If ${\varphi }_i^{-1}(u)=(f(p),f_i)$ we find ${\psi }_i^{-1}(p,u)=(p,f_i)$ where ${\psi }_i$ is the local trivialization of $f^{\ast }E$ . The transition function $t_{ij}$ at $f(p)\in U_i\cap U_j$ maps $f_j$ to $f_i=t_{ij}(f(p))f_j$ . The corresponding transition function $t_{ij}^{\ast }$ of $f^{\ast }E$ at $p\in f^{-1}(U_i)\cap f^{-1}(U_j)$ also maps $f_j$ to $f_i$ ; see figure 9.7. This shows that

Figure 9.7. The transition function $t_{ij}^{\ast }$ of the pullback bundle $f^{\ast }E$ is a pullback of the transition function $t_{ij}$ of $E$ .

$$t_{ij}^{\ast }(p)=t_{ij}(f(p)).\text{\hspace{1em}(9.18)}$$

Example 9.2

Let $M$ and $N$ be differentiable manifolds with $\dim M=\dim N=m$ . Let $f:N\to M$ be a smooth map. The map $f$ induces a map ${\pi }_2:TN\to TM$ such that the following diagram commutes:

$$\begin{array}{ccc}TN& \stackrel{{\pi }_2}{\to }& TM\\ {\pi }_1\downarrow & & \downarrow \pi \\ N& \stackrel{f}{\to }& M.\end{array}\text{\hspace{1em}(9.19)}$$

Let $W=W^{\nu }\partial /\partial y^{\nu }$ be a vector of $T_{p}N$ and $V=V^{\mu }\partial /\partial x^{\mu }$ be the corresponding vector of $T_{f(p)}M$ . If $TN$ is a pullback bundle $f^{\ast }(TM)$ , ${\pi }_2$ maps $T_{p}N$ to $T_{f(p)}M$ diffeomorphically. This is possible if and only if ${\pi }_2$ has the maximal rank $m$ at

each point of $TN$ . Let $\phi (f(p))=(f^1(y),\dots ,f^m(y))$ be the coordinates of $f(p)$ in a chart $(U,\phi )$ of $M$ , where $y=\phi (p)$ are the coordinates of $p$ in a chart $(V,\psi )$ of $N$ . The maximal rank condition is given by $\det (\partial f^{\mu }(y)/\partial y^{\nu })\ne 0$ for any $p\in N$ .

9.2.6 Homotopy axiom

Let $f$ and $g$ be maps from $M^{\prime }$ to $M$ . They are said to be homotopic if there exists a smooth map $F:M^{\prime }\times [0,1]\to M$ such that $F(p,0)=f(p)$ and $F(p,1)=g(p)$ for any $p\in M^{\prime }$ , see section 4.2.

Theorem 9.1

Let $E\stackrel{\pi }{\to }M$ be a fibre bundle with fibre $F$ and let $f$ and $g$ be homotopic maps from $N$ to $M$ . Then $f^{\ast }E$ and $g^{\ast }E$ are equivalent bundles over $N$ .

The proof is found in Steenrod (1951). Let $M$ be a manifold which is contractible to a point. Then there exists a homotopy $F:M\times I\to M$ such that

$$F(p,0)=pF(p,1)=p_0$$

where $p_0\in M$ is a fixed point. Let $E\stackrel{\pi }{\to }M$ be a fibre bundle over $M$ and consider pullback bundles $h_0^{\ast }E$ and $h_1^{\ast }E$ , where $h_t(p)\equiv F(p,t)$ . The fibre bundle $h_1^{\ast }E$ is a pullback of a fibre bundle $\{p_0\}\times F$ and hence is a trivial bundle: $h_1^{\ast }E\simeq M\times F$ . However, $h_0^{\ast }E=E$ since $h_0$ is the identity map. According to theorem 9.1, $h_0^{\ast }E=E$ is equivalent to $h_1^{\ast }E=M\times F$ , hence $E$ is a trivial bundle. For example, the tangent bundle $T{\mathbb{R}}^m$ is trivial. We have obtained the following corollary.

Corollary 9.1

Let $E\stackrel{\pi }{\to }M$ be a fibre bundle. $E$ is trivial if $M$ is contractible to a point.

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