Let $M$ be a smooth compact, oriented manifold. Let $X$ be a submanifold which is of the following type $$X := \{ p \in M: \psi(p) =0, ~~\varphi(p) \neq 0 \} $$ where $$ \psi: M \rightarrow V, \qquad \varphi: M \rightarrow W $$ are smooth sections of some oriented vector bundle $V$ and $W$. Moreover, when $\varphi(p) \neq 0$, the section $\psi$ is transverse to the zero set. In particular $X$ is a smooth oriented manifold of the right dimension. My question is the following: Given a smooth oriented vector bundle $E \rightarrow M$ such that rank of $E$ is same as dimension of $X$, is it always possible to find a smooth section $s:M \rightarrow E$, transverse to the zero set, such that $s^{-1}(0)$ intersects $X$ transversally and it does not intersect $\overline{X}-X$ anywhere, where $\overline{X}$ is the closure of $X$ inside $M$?

Secondly, is it necessarily true that the object $\overline{X}$ defines a homology class in $H_{*}(M,\mathbb{Z})$? In particular, I want to say that $$ < e(E), [\overline{X}] > = +-| X \cap s^{-1}(0) | $$

where $e(E)$ is the Euler class of $E$ and $+-|A|$ indicates the signed cardinality of the set $A$.

Note that, if $X$ was simply some smooth submanifold of $M$ then this may not have been true, because the closure could be a very large set. The point here is that $X$ is given as the vanishing and nonvanishing of some sections, which are defined on the whole of $M$. If instead $\psi$ was defined only partially on $M$ (eg a section with poles), then the statement might not have been true.


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