Transition functions and the clutching construction

Local triviality means that every property of a line bundle can be understood locally. This is like choosing co-ordinates for a manifold. Given $L \to M$ we cover $M$ with open sets $U_\alpha$ on which there are nowhere vanishing sections $s_\alpha$. If $\xi$ is a global section of $L$ then it satisfies $\xi\vert _{U_\alpha } = \xi_{\alpha} s_{\alpha}$ for some smooth $\xi_{\alpha} : U_{\alpha} \to \mathbb{C}$. The converse is also true. If we can find $\xi_{\alpha}$ such that $\xi_{\alpha} s_{\alpha} = \xi_{\beta} s_{\beta}$ for all $\alpha, \beta$ then they fit together to define a global section $\xi$ with $\xi\vert _{U_\alpha } = \xi_{\alpha} s_{\alpha}$.

It is therefore useful to define $g_{\alpha \b }: U_{\alpha} \cap U_{\beta} \to \mathbb{C}^{\times}$ by $s_{\alpha} = g_{\alpha \beta} s_{\beta}$. Then a collection of functions $\xi_{\alpha}$ define a global section if on any intersection $U_\alpha \cap U_\b$ we have $\xi_\b = g_{\alpha \b } \xi_\alpha$. The functions $g_{\alpha \b }$ are called the transition functions of $L$. We shall see in a moment that they determine $L$ completely. It is easy to show, from their definition, that the transition functions satisfy three conditions:

$$(1) \ \ \ g_{\alpha \alpha} = 1$$

$$(2) \ \ \ g_{\alpha \beta} = g_{\beta \alpha}$$

$$(3) \ \ \ g_{\alpha \beta} \ g_{\beta \gamma} \ g_{\gamma \alpha} = 1 \ \hbox{on} \ U_{\alpha} \cap U_{\beta} \cap U_{\gamma}$$

The last condition (3) is called the cocycle condition.

Proposition 1.2   Given an open cover $\{U_\alpha \}$ of $M$ and functions $g_{\alpha\beta} : U_\alpha \cap U_\beta \to \mathbb{C}^\times$ satisfying (1) (2) and (3) above we can find a line bundle $L \to M$ with transition functions the $g_{\alpha \beta}$.

Proof. Consider the disjoint union $\tilde M$ of all the $\mathbb{C}\times U_{\alpha}$. We stick these together using the $g_{\alpha \b }$. More precisely let $I$ be the indexing set and define $\tilde M$ as the subset of $I \times M$ of pairs $(\alpha , m)$ such that $m \in U_\alpha$. Now consider $\mathbb{C}\times \tilde M$ whose elements are triples $(\lambda, m, \alpha)$ and define $(\lambda, m, \alpha) \sim (\mu, n, \beta)$ if $m = n$ and $g_{\alpha \beta} (m) \lambda = \mu$. We leave it as an exercise to show that $\sim$ is an equivalence relation. Indeed ((1) (2) (3) give reflexivity, symmetry and transitivity respectively.)

Denote equivalence classes by square brackets and define $L$ to be the set of equivalence classes. Define addition by $[(\lambda, m, \alpha)] + [(\mu, m, \alpha)] = [(\lambda + \mu, m, \alpha)]$ and scalar multiplication by $z [(\lambda, m, \alpha)] = [(z\lambda, m, \alpha)].$ The projection map is $\pi ([(\lambda, m, \alpha)]) = m$. We leave it as an exercise to show that these are all well-defined. Finally define $s_{\alpha} (m) = [(1, m, \alpha)]$. Then $s_{\alpha} \ (m) = [(1, m, \alpha)] = [( g_{\alpha \beta} \ (m), m, \beta)] = g_{\alpha \beta} \ (m) s_{\beta} \ (m)$ as required.

Finally we need to show that $L$ can be made into a differentiable manifold in such a way that it is a line bundle and the $s_\alpha$ are smooth. Denote by $W_{\alpha }$ the preimage of $U_\alpha$ under the projection map. There is a bijection $\psi_{\alpha } \colon W_{\alpha } \to \mathbb{C}\times U_{\alpha }$ defined by $\psi_{\alpha }([\alpha , x, z]) = (z, x)$. This is a local trivialisation. If $(V, \phi)$ is a co-ordinate chart on $U_{\alpha } \times \mathbb{C}$ then we can define a chart on $L$ by $(\phi_{\alpha }^{-1}(V), \phi_{\alpha }\circ \psi_{\alpha })$. We leave it as an exercise to check that these charts define an atlas. This depends on the fact that $g_{\alpha \b }\colon U_{\alpha } \cap U_{\b } \to \mathbb{C}^{\times}$ is smooth. $\qedsymbol$

The construction we have used here is called the clutching construction. It follows from this proposition that the transition functions capture all the information contained in $L$. However they are by no means unique. Even if we leave the cover fixed we could replace each $s_\alpha$ by $h_{\alpha} s_{\alpha}$ where $h_{\alpha} : U_{\alpha} \to \mathbb{C}^{\times}$ and then $g_{\alpha \beta}$ becomes $h_{\alpha} g_{\alpha \beta} h_{\beta}^{-1}$. If we continued to try and understand this ambiguity and the dependence on the cover we would be forced to invent Cêch cohomology and show that that the isomorphism classes of complex line bundles on $M$ are in bijective correspondence with the Cêch cohomology group $H^{1} (M, \mathbb{C}^{\times})$. We refer the interested reader to [11,8].

Example 1.8   The tangent bundle to the two-sphere. Cover the two sphere by open sets $U_0$ and $U_1$ corresponding to the upper and lower hemispheres but slightly overlapping on the equator. The intersection of $U_0$ and $U_1$ looks like an annulus. We can find non-vanishing vector fields $s_0$ and $s_1$ as in Figure 2.

Figure 2: Vector fields on the two sphere.

If we undo the equator to a straightline and restrict $s_0$ and $s_1$ to that we obtain Figure 3.

Figure 3: The sections $s_0$ and $s_1$ restricted to the equator.

If we solve the equation $s_0 = g_{01}s_1$ then we are finding out how much we have to rotate $s_1$ to get $s_0$ and hence defining the map $g_{01} \colon U_0 \cap U_1 \to \mathbb{C}^\times$ with values in the unit circle. Inspection of Figure 3 shows that as we go around the equator once $s_0$ rotates forwards once and $s_1$ rotates backwards once so that thought of as a point on the unit circle in $\mathbb{C}^\times$ $g_{01}$ rotates around twice. In other words $g_{01} : U_{0} \cap U_1 \to \mathbb{C}^{\times}$ has winding number $2$. This two will be important latter.

Example 1.9 (Hopf bundle.)   We can define sections $s_{i} \colon U_{\alpha } \to H$ by

$$\begin{align}s_{0}[z] =& ((1, \frac{z^{1}}{ z^{0}}), [z])\\ s_{1}[z] =& (( \frac{z^{0}}{ z^{1}},1), [z]). \end{align}$$

The transition functions are

$$g_{01}([z]) = \frac{z^{1}} {z^{0}}.$$