I am trying to prove the following statement:
Let $S \subset \mathbb{R}^3$ be a compact connected surface, then $\chi(S)$ assumes one of the values $2,0,-2,...$. Additionally, if $S'\subset \mathbb{R}^3$ is another compact surface with $\chi(S)=\chi(S')$, then $S$ is homeomorphic to $S'$.
which is found in Do Carmo's "Differential Geometry of Curves and Surfaces" in the chapter on the Gauss-Bonnet theorem.
I am attempting to prove this via the Poincaré-Hopf theorem
$\sum_{\{p \in M | X(p)=0\}} ind(X,p)=\chi(M)$
My issue is, that these indices only seem to be invariant under diffeomorphisms and not homeomorphisms, which I need. Is this the wrong attempt?
I'd also be glad for any tips on how to prove the first part of the statement.