---- > [!definition] Definition. ([[orientation of a vector bundle]]) > Let $R$ be a (assume [[commutative ring|commutative]]) [[ring]]. Let $E \xrightarrow{\pi}X$ be a [[vector bundle]] of dimension $d$ with typical fiber $\mathbb{R}^{d}$. Put $E^{\sharp}:=E - s_{0}(X)$, and $E_{x}^{\sharp}=E-\{s_{0}(x)\}$ for $x \in X$. > > A **local $R$-orientation of $E$ at $x \in X$** is an $R$-[[orientation of a vector space|orientation]] of the [[vector space]] $E_{x}$, that is, a choice of $R$-[[module]] [[submodule generated by a subset|generator]] $\varepsilon_{x}$ of $H^{d}(E_{x}, E_{x}^{\sharp}; R)$. > > An **$R$-orientation of $E$** is a choice of local $R$-orientations $\{ \varepsilon_{x} \}_{x \in X}$ which are compatible in the following way: if $(U, \Phi_{U})$ is a trivializing neighborhood of $E$, and $x,y \in U$, then under the [[homeomorphism|homeomorphisms]] (also [[linear isomorphism|linear isomorphisms]])[^1]: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFEB9ADxAF9S6TLnyEUZAExVajFmy4BPfoJAZseAkQCMpLdPrNWiDgAIAPpwCqyoetFEJu-bKMhLJgDoe8AW3iePHzocAAsAIzDgACU+AD0oG1VhDTFkAGYnagM5Yy8g0IjouIS+aRgoAHN4IlAAMwAnCB8kRxAcCCQMmUM2LwAFEKwrRIampDI2jsRWnDosBjYQiAgAaxHG5sQdSZbqWfnF5bWBOo3OvamAFiyXXo80IeA84PDImPi+dbHECfaka5AYRgYASiDSE2yrhCPC+m22f0QAKBIKQAFpwTcesZoUpqAw6ECGH1kvZjPUsBUQjh+BQ+EA > \begin{tikzcd} > E_x \arrow[rd, hook] \arrow[rrrd, "h_x", bend left] & & & \\ > & E |_U \arrow[r, "\Phi_U"] & U \times \mathbb{R}^d \arrow[r, "\pi_{\mathbb{R}^d}"] & \mathbb{R}^d \\ > E_y \arrow[ru, hook] \arrow[rrru, "h_y"', bend right] & & & > \end{tikzcd} > \end{document} > ``` > [[relative singular homology|the map]] $h^{*}_{y} \circ (h_{x}^{-1})^{*}:H^{d}(E_{x}, E_{x}^{\sharp}; R) \to H^{d}(E_{y}, E_{y}^{\sharp}; R)$ > sends $\varepsilon_{x}$ to $\varepsilon_{y}$. > > If $E$ admits an $R$-orientation, then we say $E$ is **$R$-orientable**. ^definition > [!note] Remark. > This is a rather refined notion, and the strength of the condition depends on $R$. For example, any [[vector bundle]] is $\mathbb{F}_{2}$-orientable (there is only one possible choice of [[submodule generated by a subset|generator]]), while $\mathbb{Z}$-orientability is the strongest — if a [[vector bundle]] is $\mathbb{Z}$-orientable, then it is $R$-orientable for all $R$. > > The best way to check orientability is using the [[determinant criterion for orientability of a vector bundle]]. ^note ---- #### [^1]: The composition $E_{x} \hookrightarrow E |_{U} \xrightarrow{\Phi_{U}}U \times \mathbb{R}^{d} \xrightarrow{\pi_{\mathbb{R}^{d}}}\mathbb{R}^{d}$ is the restriction $\Phi_{U} |_{E_{x}}$ followed by projection onto the second coordinate. The [[vector bundle]] definition asserts that $\Phi_{U} |E_{x}$ is an [[isomorphism]] $E_{x} \to \{ x \} \times \mathbb{R}^{d}$. The subsequent projection onto $\mathbb{R}^{d}$ is also, of course, an [[isomorphism]]. ---- #### 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 > ```