# Lefschetz duality for manifolds with boundary / stratified spaces

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:

1. 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.
2. 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.
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The Lefschetz duality theorem holds for topological manifolds with boundary, so the corners aren't an issue here. Also, I'm not sure that your left-hand side is the same as the left-hand side in the Getzler--Jones paper, since faces of codimension $p+1$ might be contained in faces of codimension $p$. – Mark Grant Sep 17 '13 at 13:47
@MarkGrant I see. Thanks. So in particular a manifold with corners is a topological manifold with boundary. In the Getzler-Jones paper they write $M[p]$ for the codimension $p$ faces. So $F_p$ is the closure of $M[p]$ and in the left hand side they have $M[p]$ instead of $F_p \ F_{p+1}$. I can't see how a codimension $p+1$ face can be contained in a codimension $p$ face. Shouldn't $F_{p+1}$ be the boundary of $F_p$? I am thinking of faces being open. Eg. The open square is a codimension $0$ face of the square. – TriThang Tran Sep 17 '13 at 18:36
I guess it depends on how you define a face of a manifold with corners. There are competing definitions, I think, which are discussed in the article of Joyce arxiv.org/abs/0910.3518. In any case, the duality homomorphism should reduce to Lefschetz duality for manifolds with boundary (when $p=0$), as these are a special case of manifolds with corners. – Mark Grant Sep 17 '13 at 21:32