# Relation between the orientation sheaves of the interior and the boundary of a topological manifold

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 $$D^{b}(\partial M)$$?