Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus \partial M$ and $\partial M$ with respect to some fixed field $k$, respectively. Let $i \colon M \setminus \partial M \rightarrow M$ and $j \colon \partial M \rightarrow M$ denote the inclusions. Then, are $j^{\ast}R i_{\ast} \mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ isomorphic in the bounded derived category $D^{b}(\partial M)$?



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