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Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset X$ is a closed submanifold of a closed manifold $X$ (dim$X=n$, codim$Y=k$) with orientable normal bundle then the Leray sequence is the exact sequence \begin{equation*} \cdots \to H_i(X-Y) \to H_i(X) \to H_{i-k}(Y) \to H_{i-1}(X-Y) \to \cdots \end{equation*} Thus, Euler characteristics of these spaces satisfy $\chi(X) = \chi(X-Y)+\chi(Y)$. How is this interpreted in terms of the Poincaré–Hopf theorem that says the Euler characteristic of a closed manifold $M$ is \begin{equation*} \sum_a \mbox{index}_{x_i}(v) = \chi(M) \end{equation*} where the sum is over isolated zeroes of a vector field $v \in TM$? Can the intuitive argument that the sum of the right hand side decomposes into two parts (on and off $Y$) simply be used to prove additivity of the Euler characteristic in this case? It seems too simple to be true...

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    $\begingroup$ You should probably use homotopy invariance and extend Poincare theorem to manifolds with boundary. But this is not a research question, so I'm voting to close. $\endgroup$ Feb 4, 2016 at 16:33
  • $\begingroup$ Why is this not a research question? $\endgroup$
    – Tobias
    Feb 4, 2016 at 16:38
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    $\begingroup$ Don't even know what to say. It looks like an algebraic topology homework. In fact, I would give a question like this as a homework. $\endgroup$ Feb 4, 2016 at 16:48

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