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Let $\mathbb{R}^2\to E\to B$ be a 2-dimensional real vector bundle. Consider:

  • The fibrewise Alexandroff compactification $(\mathbb{S}^2,\infty)\to S\to B$
  • The $\infty$-section $s_\infty:B\to S$ and $B_\infty:=s_\infty(B)\subseteq S$
  • The zero section $s_0:B\to E\subseteq S$.
  • The inclusion $\imath:S\to (S,B_\infty)$, the Thom class $\tau\in H^2(S,B_\infty)$ and $\chi:=\imath^*\tau\in H^2(S)$.

We know that $e:=s_0^*\chi\in H^2(B)$ is the Euler class. Now consider a section $s:B\to S$. The question is what we can say about $s^*\chi$. The interesting set seems to be $D:=s^{-1}(\infty)\subseteq B$. Clear is:

  1. If $D=\emptyset$, then $s\simeq s_0$ and $s^*\chi=e$.
  2. If $D=B$, then $s=s_\infty$ and $s^*\chi=0$.

What about other cases? We may for my purposes assume that $D$ is a »nice« subspace. It seems that the Euler class of $E|_D\to D$ has to be taken into account.

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  • $\begingroup$ Okay, I found a partial answer: If $D$ is a smooth submanifold of codimension $d\ge 3$, then $H^2(E\setminus D\to E)$ should be injective (Thom isomorphism) and $s^*\chi=e$. But for other cases? $\endgroup$
    – FKranhold
    Commented Apr 23, 2018 at 11:36

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