For this question let $k$ be any field of characteristic not equal to $2$ and $X$ a stratified space whose strata are topological manifolds. I'm not sure what the definition of "stratified space" should be here, but right now I only care about the case that $X$ is a solid (filled-in) regular $n$-gon in the plane.

What is an appropriate definition of a $k$-orientation in this situation (hopefully one in terms of sheaf cohomology)? I understand how to do this for a manifold with boundary, but there are some singularities. Also, how does this relate to the dualizing complex? There is a rather cryptic remark in Gelfand and Manin where they claim that the cohomology sheaves of the dualizing complex are constructible with respect to the given stratification.

Sorry for the disorganized and possibly confused question. I would be happy with a reference, if anyone knows of one. By the way, I'm using the rt.representation-theory tag because this relates to the Weil representation and I couldn't think of another arXiv subject tag.