Hu Hanyang

Hu Hanyang

Master's student in ECE at UC San Diego, specializing in Intelligent Systems, Robotics & Control.

Uniqueness of Vector Bundles From a Collection of Transition Functions

Reference. Raymond O. Wells, Differential Analysis on Complex Manifolds, Chapter 1.2, Graduate Texts in Mathematics 65, Springer (2008).

Construction

Let $X$ be an $\mathcal{S}$-manifold with an open cover $X=\bigcup_{\alpha\in I}U_\alpha$. Whenever $U_\alpha\cap U_\beta\neq\varnothing$, suppose that we have an $\mathcal{S}$-map \[g_{\alpha\beta}: U_\alpha \cap U_\beta \to \text{GL}(r, K)\] satisfying the cocycle conditions:

  1. $g_{\alpha\beta}(q) \circ g_{\beta\gamma}(q) \circ g_{\gamma\alpha}(q) = \mathbf{I}r$ for $q \in U\alpha \cap U_\beta \cap U_\gamma$;

  2. $g_{\alpha\alpha}(q) = \mathbf{I}{r}$ for $q \in U\alpha$.

Consider the disjoint union $\widetilde E=\bigsqcup_{\alpha\in I}U_\alpha\times K^r$ with its natural product topology and $\mathcal{S}$-structure. For $(x,v)\in U_\beta\times K^r$ and $(y,w)\in U_\alpha\times K^r$, define $(x,v)\sim(y,w)$ if and only if $y=x$ and $w=g_{\alpha\beta}(x)v$. The cocycle conditions make this an equivalence relation.

We have a natural surjection $\text{CL}: \tilde{E} \to E$ by sending $(x, v)$ to its equivalence class. We equip $E$ with the quotient topology, then $\pi: E \to X$ defined by $\pi(\text{CL}(x, v)) = x$ is an $\mathcal{S}$-bundle.

This proves existence. A classmate in MA5210 Differentiable Manifolds then asked whether the construction is unique. Let $\pi_0:E_0\to X$ and $\pi_1:E_1\to X$ be $\mathcal{S}$-bundles with the same trivializing cover and transition functions. They are indeed isomorphic as $\mathcal{S}$-bundles.

Uniqueness up to bundle isomorphism

My approach was inspired by a remark in [Wells] (after Example 2.12) on how to construct a new section by putting together a collection of compatible sections defined on trivializing open sets. In general, this is the philosophy of sheaves.

Let $\phi_\alpha: \pi_0^{-1}(U_\alpha) \to U_\alpha \times K^r$ and $\psi_\alpha: \pi_1^{-1}(U_\alpha) \to U_\alpha \times K^r$ be local trivializations of $\pi_0$ and $\pi_1$ respectively. Define $f: E_0 \to E_1$ such that \[f(v) = (\psi_\alpha^{-1} \circ \phi_\alpha)(v) \in \pi_1^{-1}(U_\alpha) \subset E_1\] for all $v \in \pi_0^{-1}(U_\alpha)$. We shall verify that $f$ is well-defined.

For all $v \in \pi_0^{-1}(U_\alpha) \cap \pi_0^{-1}(U_\beta) \not= \emptyset$, let $(x, w) = \phi_\beta(v)$ where $v \in E_{0,x}$ and $w \in K^r$, then \[(\psi_\alpha\circ\psi_\beta^{-1})(\phi_\beta(v)) = (\psi_\alpha\circ\psi_\beta^{-1})(x, w) = (x, g_{\beta\alpha}(x)w) = \phi_\alpha(v).\] Consequently, we have $(\psi_\alpha^{-1} \circ \phi_\alpha)(v) = (\psi_\beta^{-1} \circ \phi_\beta)(v)$, i.e. $f$ is indeed well-defined.

It is easy to verify that $f: E_0 \to E_1$ is bijective (the inverse is $f^{-1}(v) = (\phi_\alpha^{-1}\circ \psi_\alpha)(v)$ for $v \in \pi_1^{-1}(U_\alpha)$) and is an $\mathcal{S}$-isomorphism (since $\phi_\alpha, \psi_\alpha$ are $\mathcal{S}$-isomorphisms).

Furthermore, $f$ is fibre-preserving and gives a $K$-linear isomorphism on each fibre. Specifically, for every $p \in U_\alpha \subset X$, we have \[f_p: E_{0,p} \xrightarrow{\phi_\alpha} \{p\}\times K^r \xrightarrow{\psi_\alpha^{-1}} E_{1,p}.\] Consequently, $f: E_0 \to E_1$ is an $\mathcal{S}$-bundle isomorphism.

Reflection

This argument resembles the gluing principle for sheaves: compatible local objects determine a global object. It also sits behind the correspondence between vector bundles and locally free sheaves, although establishing that equivalence requires more than the uniqueness argument given here.