Let $M$ be a manifold with corners. Let $F_p$ denote the union of all the codimension $i \geq p$ faces of $M$. Then I have read that there is a form of Lefschetz duality that says that there is an isomorphism $$H_q(F_p \setminus F_{p+1}) \cong H^{n-p-q}(F_p, F_{p+1}).$$

(This was in the "Operads and Homotopy Algebra" paper of Getzler-Jones, though I assume that this fact is well known).

I have two questions:

- The usual Lefschetz duality that I am familiar with is for manifolds with boundary. It says $H_*(M, \partial M) \cong H^{\dim M - *}(M)$. Is the above isomorphism just this one in disguise, taking $F_{p+1}$ as the boundary of $F_p$? What is happening to the rest of the "corners"? It is also the case that $F_p$ is not a manifold as far as I can tell... it is just one with corners.
- If instead $M$ was a stratified space such that the $F_p$ were the codimension $i \geq p$ strata, with $F_p \setminus F_{p+1}$ a manifold, then does the above isomorphism hold? Here that strata $F_p \setminus F_{p+1}$ may be empty.