A nowhere-zero section forces the Euler class to vanish

The section can be normalised. Choose a bundle metric on the numerable bundle $E$ by [F2] and put $\widehat{\sigma }(b)=\sigma (b)\mathrm{/}\mathrm{\parallel }\sigma (b)\mathrm{\parallel }$; this is continuous because $\sigma$ is nowhere zero, and it is a section of the sphere bundle $p:S(E)\to B$ in the sense of [F4], that is, $p\widehat{\sigma }={\mathrm{id}}_B$. The normalisation is a specified function of the supplied section and metric, not a choice.

The sphere projection is injective on cohomology. Since $p\widehat{\sigma }={\mathrm{id}}_B$, functoriality [F3] gives ${\widehat{\sigma }}^{\ast }{p}^{\ast }=({\mathrm{id}}_B{)}^{\ast }=\mathrm{id}$ on $H^{\ast }(B;R)$; a map with a left inverse is injective, so $p^{\ast }:H^k(B;R)\to H^k(S(E);R)$ is injective for every $k$.

The Euler class vanishes. Take $k=n$ in the Gysin sequence of [F1]: $H^0(B;R)\stackrel{⌣e(E)}{\to }{H}^n(B;R)\stackrel{p^{\ast }}{\to }{H}^n(S(E);R)\mathrm{.}$ Exactness at $H^n(B;R)$ says that the kernel of $p^{\ast }$ is the image of $⌣e(E)$. Whether or not $B$ is connected, the particular element $e(E)=1⌣e(E)$ lies in this image. By step 2.1 the kernel is zero, so $e(E)=0$. No assertion that the whole image is cyclic is needed.

Boundary cases and the missing converse. The hypothesis $n\ge 1$ is used in two places: the sphere bundle has nonempty fiber $S^{n-1}$, and the unit normalisation of step 1.1 divides by the positive norm of a nonzero vector of a positive-dimensional fiber. For $n=0$ every section is the zero section, and the statement is excluded; with the standard unit orientation its Euler class is $e(0_B,1)=1$, while an arbitrary supplied rank-zero orientation $o$ gives $e(0_B,o)=o$. If $\sigma$ is the zero section of a positive-rank bundle the normalisation is undefined, and indeed the conclusion can fail. The proposition asserts no converse: the vanishing of $e(E)$ does not in general produce a nowhere-zero section, and no such section is constructed here. Over the empty base the unique class is zero. AC is used through the metric of [F2] and the Gysin sequence, as recorded.