Uniqueness of Vector Bundles From a Collection of Transition Functions
Reference. Chapter 1.2 of [Wells] Raymond O. Wells, Differential Analysis on Complex Manifolds. Graduate Texts in Mathematics Vol. 65, Springer (2008).
We could construct vector bundles from a collection of transition functions:
Let \(X\) be an \(\mathcal{S}\)-manifold. Suppose \(X\) has an open covering \(X = \cup_{\alpha \in I} U_\alpha\) and whenever \(U_\alpha \cap U_\beta \not= \emptyset\), we have an \(\mathcal{S}\)-funcion \[g_{\alpha\beta}: U_\alpha \cap U_\beta \to \text{GL}(r, K)\] satisfying the following equalities:
\(g_{\alpha\beta}(q) \circ g_{\beta\gamma}(q) \circ g_{\gamma\alpha}(q) = \mathbf{I}_r\) for \(q \in U_\alpha \cap U_\beta \cap U_\gamma\);
\(g_{\alpha\alpha}(q) = \mathbf{I}_{r}\) for \(q \in U_\alpha\).
Consider the disjoint union \(\tilde{E} = \sqcup_{\alpha \in I} U_\alpha \times K^r\) equipped with the natural product topology and \(\mathcal{S}\)-structure. Define an equivalence relation on \(\tilde{E}\) such that for \((x, v) \in U_\beta \times K^r\) and \((y, w) \in U_\alpha \times K^r\), \((x, v) \sim (y, w)\) iff. \(y = x\) and \(w = g_{\alpha\beta}(x)v\). This is indeed an equivalence relation due to the constraints on the transition functions.
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 shows the existence of vector bundles given a collection of transition functions, and it is natural to ask whether such construction is unique (which is a question raised by my classmate in MA5210 Differentiable Manifolds). Let \(\pi_0: E_0 \to X\) and \(\pi_1: E_1 \to X\) be \(\mathcal{S}\)-bundles with the same trivializing open cover \(\{U_\alpha\}\) and collection of transition functions \(\{g_{\alpha\beta}\}\), it turns out that they are indeed \(\mathcal{S}\)-bundle isomorphic.
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.
Thoughts. Can we connect this to the one-one correspondence between vector bundles and locally free sheaves? I guess no. Although the proof that every locally free sheaves gives us a vector bundle uses the sheaf maps to construct a collection of transition maps, we only need the existence instead of the uniqueness.