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Recall, that if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$.

Is there an analogous statement for constructible sheaves (say of $\mathbb C$-modules) on a (real) manifold? Specifically, if $\mathcal F$ is constructible on $\mathbb R^n$ and $i$ is the embedding $\mathbb R^{n-1} \hookrightarrow \mathbb R^n$, does $H^j(i^! \mathcal F)$ vanish for $j > 1$?

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