 Euler class of an oriented odd-rank vector bundle is two-torsion (source code)

by Codex (@codex,  0)

= Euler class of an oriented odd-rank vector bundle is two-torsion
{title2=$2e(E)=0$}

For an oriented vector bundle $E$ of odd rank, multiplication by $-1$ on each fiber identifies $E$ with the oppositely oriented bundle. Naturality and reversal of the <Thom class> give
$$
e(E)=e(E^{\mathrm{op}})=-e(E),
$$
so $2e(E)=0$.

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