----- > [!proposition] Proposition. ([[determinant criterion for orientability of a vector bundle]]) > Let $E \xrightarrow{\pi}X$ be a [[vector bundle]] of dimension $d$ with typical fiber $\mathbb{R}^{d}$. > Let $\{ U_{\alpha} \}$ be a trivializing [[cover]] of $E$. If the [[transition functions for a vector bundle over a smooth manifold|transition functions]] determined by the $U_{\alpha}$ are [[orientation-preserving and orientation-reversing operators|orientation-preserving]], then $E$ is $R$-[[orientation of a vector bundle|orientable]]. Explicitly, if we have for all $\alpha,\beta \in I$ that $(U_{\alpha} \cap U_{\beta}) \times \mathbb{R}^{d} \xrightarrow[\cong]{\Phi_{U_{\alpha}} ^{-1}} E |_{U_{\alpha} \cap U_{\beta}} \xrightarrow[\cong]{\Phi_{U_{\beta}}} (U_{\alpha} \cap U_{\beta}) \times \mathbb{R}^{d}$ has positive [[determinant]] on each fiber, then $E$ is orientable for *any* [[ring]] $R$.[^1] > Note that we don't have to check the [[determinant]] at each point on $U_{\alpha} \cap U_{\beta}$. By [[continuous|continuity]], we only have to check it for one point. > [!proof]- Proof. ([[determinant criterion for orientability of a vector bundle]]) > Choose any generator $u \in H^{d}(\mathbb{R}^{d}, \mathbb{R}^{d}-\{ 0 \};R)$. Then for $x \in U_{\alpha}$, we define a generator $\varepsilon_{x}$ of $H^{d}(E_{x}, E_{x}^{\sharp}; R)$ by [[relative singular homology|pulling back]] $u$ along $E_{x} \hookrightarrow E_{U_{\alpha}} \xrightarrow{\Phi_{\alpha}} U_{\alpha} \times \mathbb{R}^{d} \to \mathbb{R}^{d} \ \ ({\dagger}_{\alpha})$ (viewed as a [[topological pair|map of pairs]] $(E_{x}, E_{x}^{\sharp}) \to (\mathbb{R}^{d}, \mathbb{R}^{d}-\{ 0 \})$, and as a [[linear isomorphism]]). If $x \in U_{\beta}$ as well, then the analogous [[linear isomorphism]] ${\dagger}_{\beta}$ differs from ${\dagger}_{\alpha}$ by post-composition with [[linear map]] $L:\mathbb{R}^{d} \to \mathbb{R}^{d}$ of positive determinant, by the assumption. > > ![[Pasted image 20250517124853.png]] > > Now recall that any linear map of positive determinant is [[homotopy|homotopic]] to the [[identity map]]. Indeed, $\text{GL}^{+}_{d}(\mathbb{R})$ is a [[path-connected]] [[group]], and a path between them will give a [[homotopy]] between the maps they represent. So we know ${\dagger}_{\alpha}$ and ${\dagger}_{\beta}$ are homotopic, and [[homotopy invariance of singular homology|hence induce]] the same map on [[singular cohomology|cohomology classes]]. ----- #### [^1]: With notation as in [[transition functions for a vector bundle over a smooth manifold|transition function]], this is saying that outputs of [[general linear group|the map]] $\varphi_{\alpha \beta}:U_{\alpha} \cap U_{\beta} \to \text{GL}(\mathbb{R}^{n})$ always have positive [[determinant]]. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```