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So I am trying to prove this, as an excercise for a course in differential topology, the previous part of the excercise was to find $\chi(S^n)$, which is $1+(-1)^n$, it is likely that this fact is meant to use in this part.

I really do not know how to begin, and all proofs I have found, use algebraic topology.

Any help would be appreciated

Bajo Fondo
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  • What definition of Euler characteristic do you use? –  Dec 14 '18 at 21:44
  • We just define the Euler characteristic as the Index sum at the zeros of a vector field in $M$, a compact differentiable manifold. (Our Poincare-Hopf just tells you that the index sum is vector field invariant). – Bajo Fondo Dec 14 '18 at 21:50

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HINT: Let $\vec v$ be a "good" vector field on your compact manifold $M$. (So it has isolated zeroes, each with index $\pm 1$, although we really don't need this.) If $p$ is a zero of $\vec v$ with index $k$, what is the index of $-\vec v$ at $p$?

Ted Shifrin
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  • Ok.. I get it... Since $v$ and $-v$ share the same zeros you have that if $v$ has index $k$ at $p$, then $deg(dg^{-1}ovog)$ is $k$ ($g:U \to V \subset M$ parametrization). Then the index of $-v$ at $p$ is $deg(dg^{-1}o-vog)=deg(-dg^{-1}ovog)$. Take $f(x)=-x$, then the index of $-v$ is $deg(fo(dg^{-1}o-vog))=deg(f)deg(dg^{-1}o-vog)=(-1)^nk=-k$. Then add up all of then and you will have that $sum \iota = -\sum \iota$, hence $\sum \iota =0$. Thank you. – Bajo Fondo Dec 14 '18 at 23:40
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    Looks right, except that you need to learn some MathJax :) – Ted Shifrin Dec 14 '18 at 23:51