# Euler Class Definition (Spivak)

In the Spivak's book on Differential Geometry Euler class $$\chi(\xi)$$ of an oriented $$k-$$plane bundle $$\xi=\pi:E\to M$$ is defined as $$s^*[\omega]\in H^k(M)$$ where $$s:M\to E$$ is any section and $$[\omega]\in H^k_c(E)$$ the Thom class represented by $$\omega$$ closed $$k$$-form with compact support on $$E$$.

Now for a non-zero section $$s$$, if we pick $$c>0$$ sufficiently large, some subset of image of $$M$$ under the section $$c\cdot s$$ will lie outside of support $$\omega$$. So by definition $$\chi(\xi)=(c\cdot s)^*[\omega]=0 \in H^k(M).$$ But why is it zero?

For example $$\omega_{cs(p)}$$ will be zero form for some $$p\in M$$. Is the form then exact?

• Perhaps you want a nowhere zero section instead of a nonzero section. If such a section exists, the eular class of the vector bundle is zero. Mar 31, 2020 at 19:15
• @AGlearner Could you perhaps elaborate on why should it be zero in that case. Thank you! Mar 31, 2020 at 19:21
• It is exactly what you wrote, if a section $s$ does not vanish, by scaling it will miss the support of the $\omega$, so pullback is a zero form. Mar 31, 2020 at 19:26
• And note that $M$ is compact? Mar 31, 2020 at 22:54
• @TedShifrin Yes :) Mar 31, 2020 at 23:01