----- > [!proposition] Proposition. ([[Euler class obstructs existence of nonvanishing section]]) > - Let $X$ be a [[topological space]] and $E \xrightarrow{\pi} X$ be a [[vector bundle]]. Put $E^{\sharp}:=E - s_{0}(X)$. > - Let $R$ be a ([[commutative ring|commutative]]) [[ring]]. > - Denote by $e(E)=s_{0}^{*}q^{*}u_{E}\in H^{d}(X;R)$ the [[The Thom isomorphism theorem|Euler class]] of $E$, where $u_{E} \in H^{d}(E, E^{\sharp};R)$ is the [[The Thom isomorphism theorem|Thom class]] of $E$. > > > If there is a [[section]] $s:X \to E$ which is nowhere zero, then $e(E)=0 \in H^{d}(X;R)$. > [!proposition] Corollary. > Contrapositively, if $e(E)$ is nontrivial, then every section of $\pi$ vanishes somewhere. In particular, $E$ must be nontrivial — $E \neq X \times \mathbb{R}^{d}$ — whenever $e(E)$ is. > > (Though $e(E)$ is 'often' trivial. For example, $e(E)=0$ [[the Euler class of an odd-rank oriented vector bundle is torsion|whenever]] $H^{d}(X;R)$ has $2$-[[torsion element of a module|torsion]].) ^proposition > [!proof]- Proof. ([[Euler class obstructs existence of nonvanishing section]]) > Assume $s:X \to E$ is nowhere zero, so that $s$ restricts to a map $X \to E^{\sharp}$. > > Consider [[long exact sequence for relative singular homology|long exact sequence for the pair]] $(E,E^{\sharp})$, giving the diagram > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAJAPSgAoBRUgAJ+XADpi4AC3oAnNAG4ASgEoQAX1LpMufIRTkK1Ok1btuffsrWbt2PASIAmIzQYs2iTjwHjJM+QkhVQ0tEAx7PSJDJ2N3My8LXgANa1C7XUcUMmI4008QZgB9fnTwnQd9ZABmUly3fPZAFAIyiMyqw2q8j3YYARtjGCgAc3giUAAzWQgAWyQyEBwIJEMTHq8ARy4AKjKp2ZWaJaQXNYSQCXwceh296bnEVePEWrOCuCLiW9sQfYfT56veLvb5hP5IAAsR2WiAArDQpDB6FB2JAwGwaIwsOj2FAIDgcEM7gc4dCkAA2GgAIxgYBRLwWWJxXjxBKJ6ko6iAA > \begin{tikzcd} > u_E \arrow[rrr, no head, dotted] & & & — \arrow[llddd, dotted, bend left] \\ > {H^d(E, E^\sharp;R)} \arrow[r, "q^*"] & H^d(E;R) \arrow[r, "\iota^*"] \arrow[d, "s_0^*"] & H^d(E^\sharp\ R) \arrow[ld, "s^*"] & \\ > & H^d(X;R) & & \\ > & e(E) & & > \end{tikzcd} > \end{document} > ``` > > which commutes because $s_{0},s$ are [[homotopy|homotopic]] (as all sections are) thus [[singular (co)chain map and homomorphism induced by a continuous map|induce]] the same maps $s_{0}^{*},s^{*}:H^{d}(E;R) \to H^{d}(X;R)$ on [[singular cohomology|cohomology]]. The top row is [[exact sequence|exact]], meaning that $u_{E} \in H^{d}(E, E^{\sharp};R)$ dies by the time it gets to $H^{d}(E^{\sharp};R)$, and so $s ^{*}$ sends it to $0 \in H^{d}(X;R)$. But this is precisely the Euler class $s_{0}^{*}q^{*}u_{E}=e(E)$. So the Euler class vanishes. ----- #### ---- #### 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 > ```