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You make several assumptions, one being that $X$ is a stratified space, carrying an orientation class. The answer to you question is yes, under more restrictive assumptions. Here they are.

• The section $s$ vanishes transversally along $M$.
• The zero set $s^{-1}(0)$ intersects the stratified space $X$ transversally.

The complete rigorous proof is a bit more involved, and the clearest argument I know is sheaf theoretic, and it involves a sheaf theoretic version of the Poincare duality, also known as Verdier duality. All the information you need you can find in B. Iversen's book Cohomology of Shaves, especially chapters IX and X.

Addendum I realize you do not need these stringent conditions. Denote by $V_X$ the restriction of $V$ to $X$ and by $\tau_X$ its Thom class viewed as a class in the local cohomology of $V_X$ along $X$, $\tau_X\in H^k_X(V_X)$ (integer coefficients). We can then arrange that $\tau_X$ has support in a tiny neighborhood of $X$ in $V_X$. Then $e(V_X)=s^*\tau_X\in H^k(X)$ is supported in a tiny open neighborhood $N$ of $s^{-1}(0)\cap X$ in $X$, i.e., $e(V_X)$ is in the image of $H^k(X, X\setminus N)$ in $H^k(X)$. Now use the technology in Iversen to conclude that

$$ \langle e(V_X), [X]\rangle =\sum_{s(x)=0} \epsilon(x), $$

where $\epsilon(x)\in\{\pm 1\}$ is the local Euler number of $S$ at $x$, and $\langle-,-\rangle $ denotes the pairing between cohomology and homology. (This is a bit long.)

You make several assumptions, one being that $X$ is a stratified space, carrying an orientation class. The answer to you question is yes, under more restrictive assumptions. Here they are.

• The section $s$ vanishes transversally along $M$.
• The zero set $s^{-1}(0)$ intersects the stratified space $X$ transversally.

The complete rigorous proof is a bit more involved, and the clearest argument I know is sheaf theoretic, and it involves a sheaf theoretic version of the Poincare duality, also known as Verdier duality. All the information you need you can find in B. Iversen's book Cohomology of Sheaves, especially chapters IX and X.

Addendum I realize you do not need these stringent conditions. Denote by $V_X$ the restriction of $V$ to $X$ and by $\tau_X$ its Thom class viewed as a class in the local cohomology of $V_X$ along $X$, $\tau_X\in H^k_X(V_X)$ (integer coefficients). We can then arrange that $\tau_X$ has support in a tiny neighborhood of $X$ in $V_X$. Then $e(V_X)=s^*\tau_X\in H^k(X)$ is supported in a tiny open neighborhood $N$ of $s^{-1}(0)\cap X$ in $X$, i.e., $e(V_X)$ is in the image of $H^k(X, X\setminus N)$ in $H^k(X)$. Now use the technology in Iversen to conclude that

$$ \langle e(V_X), [X]\rangle =\sum_{s(x)=0} \epsilon(x), $$

where $\epsilon(x)\in\{\pm 1\}$ is the local Euler number of $S$ at $x$, and $\langle-,-\rangle $ denotes the pairing between cohomology and homology. (This is a bit long.)

You make several assumptions, one being that $X$ is a stratified space, carrying an orientation class. The answer to you question is yes, under more restrictive assumptions. Here they are.

• The section $s$ vanishes transversally along $M$.
• The zero set $s^{-1}(0)$ intersects the stratified space $X$ transversally.

The complete rigorous proof is a bit more involved, and the clearest argument I know is sheaf theoretic, and it involves a sheaf theoretic version of the Poincare duality, also known as Verdier duality. All the information you need you can find in B. Iversen's book Cohomology of Shaves, especially chapters IX and X.

Addendum I realize you do not need these stringent conditions. Denote by $V_X$ the restriction of $V$ to $X$ and by $\tau_X$ its Thom class viewed as a class in the local cohomology of $V_X$ along $X$, $\tau_X\in H^k_X(V_X)$ (integer coefficients. We can then arrange that $\tau_X$ has support in a tiny neighborhood of $X$ in $V_X$. Then $e(V_X)=s^*\tau_X\in H^k(X)$ is supported in a tiny open neighborhood $N$ of $s^{-1}(0)\cap X$ in $X$, i.e., $e(V_X$ is in the image of $H^k(X, X\setminus N)$ in $H^k(X)$. Now use the technology in Iversen to conclude that

$$ \langle e(V_X), [X]\rangle =\sum_{s(x)=0} \epsilon(x), $$

where $\epsilon(x)\in\{\pm 1\}$ is the local Euler number of $S$ at $x$, and $\langle-,-\rangle $ denotes the pairing between cohomology and homology. (This is a bit long.)

You make several assumptions, one being that $X$ is a stratified space, carrying an orientation class. The answer to you question is yes, under more restrictive assumptions. Here they are.

• The section $s$ vanishes transversally along $M$.
• The zero set $s^{-1}(0)$ intersects the stratified space $X$ transversally.

The complete rigorous proof is a bit more involved, and the clearest argument I know is sheaf theoretic, and it involves a sheaf theoretic version of the Poincare duality, also known as Verdier duality. All the information you need you can find in B. Iversen's book Cohomology of Shaves, especially chapters IX and X.

Addendum I realize you do not need these stringent conditions. Denote by $V_X$ the restriction of $V$ to $X$ and by $\tau_X$ its Thom class viewed as a class in the local cohomology of $V_X$ along $X$, $\tau_X\in H^k_X(V_X)$ (integer coefficients). We can then arrange that $\tau_X$ has support in a tiny neighborhood of $X$ in $V_X$. Then $e(V_X)=s^*\tau_X\in H^k(X)$ is supported in a tiny open neighborhood $N$ of $s^{-1}(0)\cap X$ in $X$, i.e., $e(V_X)$ is in the image of $H^k(X, X\setminus N)$ in $H^k(X)$. Now use the technology in Iversen to conclude that

$$ \langle e(V_X), [X]\rangle =\sum_{s(x)=0} \epsilon(x), $$

where $\epsilon(x)\in\{\pm 1\}$ is the local Euler number of $S$ at $x$, and $\langle-,-\rangle $ denotes the pairing between cohomology and homology. (This is a bit long.)

You make several assumptions, one being that $X$ is a stratified space, carrying an orientation class. The answer to you question is yes, under more restrictive assumptions. Here they are.

• The section $s$ vanishes transversally along $M$.
• The zero set $s^{-1}(0)$ intersects the stratified space $X$ transversally.

The complete rigorous proof is a bit more involved, and the clearest argument I know is sheaf theoretic, and it involves a sheaf theoretic version of the Poincare duality, also known as Verdier duality. All the information you need you can find in B. Iversen's book Cohomology of Shaves, especially chapters IX and X.

You make several assumptions, one being that $X$ is a stratified space, carrying an orientation class. The answer to you question is yes, under more restrictive assumptions. Here they are.

• The section $s$ vanishes transversally along $M$.
• The zero set $s^{-1}(0)$ intersects the stratified space $X$ transversally.

The complete rigorous proof is a bit more involved, and the clearest argument I know is sheaf theoretic, and it involves a sheaf theoretic version of the Poincare duality, also known as Verdier duality. All the information you need you can find in B. Iversen's book Cohomology of Shaves, especially chapters IX and X.

Addendum I realize you do not need these stringent conditions. Denote by $V_X$ the restriction of $V$ to $X$ and by $\tau_X$ its Thom class viewed as a class in the local cohomology of $V_X$ along $X$, $\tau_X\in H^k_X(V_X)$ (integer coefficients. We can then arrange that $\tau_X$ has support in a tiny neighborhood of $X$ in $V_X$. Then $e(V_X)=s^*\tau_X\in H^k(X)$ is supported in a tiny open neighborhood $N$ of $s^{-1}(0)\cap X$ in $X$, i.e., $e(V_X$ is in the image of $H^k(X, X\setminus N)$ in $H^k(X)$. Now use the technology in Iversen to conclude that

$$ \langle e(V_X), [X]\rangle =\sum_{s(x)=0} \epsilon(x), $$

where $\epsilon(x)\in\{\pm 1\}$ is the local Euler number of $S$ at $x$, and $\langle-,-\rangle $ denotes the pairing between cohomology and homology. (This is a bit long.)