----- > [!proposition] Proposition. ([[the Euler class of an odd-rank oriented vector bundle is torsion]]) > - Let $X$ be a [[topological space]] and $E \xrightarrow{\pi} X$ be an [[orientation of a vector bundle|oriented]] [[vector bundle]] of rank $d$. 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 $d$ is *odd*, then the [[The Thom isomorphism theorem|Euler class]] of $E$ is [[torsion element of a module|torsion]]. Specifically, $2e(E)=0 \in H^{d}(X;R).$ > [!justification] > This is a disappointing result, because if we already know that $H^{d}(X;R)$ has no $2$-torsion (e.g. if it is free, like it is when $R$ is a [[field]]) then $e(E)=0$. For example, the Euler class of any odd-rank orientable vector bundle on $\mathbb{S}^{n}$ or $T^{n}$ is zero. ^justification > [!proof]- Proof. ([[the Euler class of an odd-rank oriented vector bundle is torsion]]) > Summary: > - consider antipodal map $a$, induces a reversal of $\varepsilon_{x}$ in each fiber > - Use uniqueness of Thom class $u_{E}$ to see $a^{*}(u_{E})=-u_{E}$. > - Argue $s_{0}^{*}a^{*}u_{E}=s_{0}^{*}u_{E}$, and finish from here > > Consider the '[[antipodal map]]' $a:E \to E$ given by negation on each fiber. It [[singular (co)chain map and homomorphism induced by a continuous map|induces]] an [[isomorphism]] $a^{*}:H^{d}(E, E^{\sharp}; R) \xrightarrow{\cong}H^{d}(E, E^{\sharp}; R)$ > given by multiplication by $(-1)^{d}=(-1)$[^1] on each '$(d-1)$-sphere' $H^{d}(E_{x}, E_{x}^{\sharp};R)$. So $a^{*}u_{E}$ restricts to $-\varepsilon_{x}$ in each fiber $E_{x}$. Therefore, by uniqueness of $u_{E}$, $a^{*}(u_{E})=-u_{E}.$We also know $a \circ s_{0}=s_{0}$, which implies $s_{0}^{*}\big( a^{*}(u_{E}) \big)=s_{0}^{*}(u_{E}).$ > Combining these two facts: $s_{0}^{*}(-u_{E})=s_{0}^{*}(u_{E}),$ > i.e., $2s_{0}^{*}(u_{E})=0$. (I think we suppressed $q^{*}$ from the notation, but it all makes sense. If we did not suppress $q^{*}$ then I am confused.) ----- #### [^1]: Recall the analogous theory for $\mathbb{S}^{n}$. Stuff like [[antipodal map on the sphere is homotopic to the identity in odd dimensions]] [[degree of the antipodal map]] [[map on the sphere with no fixed points is homotopic to antipodal map]] stuff like that; importantly realizing $a$ is composition of reflections each having degree $-1$ [[the degree of a reflection is -1]]. ---- #### 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 > ```