8
$\begingroup$

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me.

The exact statement in question is: Let $X$ be a Hausdorff topological space, and let $A \subset X$ be a subspace such that the complement $X - A$ is an orientable topological $n$-manifold. Then, for any abelian group $G$ and any $i$, $$H_i(X, A; G) \cong H_c^{n - i}(X - A; G),$$ where $H_i$ and $H_c^{n - i}$ denote singular homology and compactly supported singular cohomology, respectively.

The problem is, by Poincaré duality, $H_c^{n - i}(X - A; G) \cong H_i(X - A; G)$, so if the above statement is true, then $H_i(X - A; G) \cong H_i(X, A; G)$. But that definitely doesn't seem right; for example, if $X = \mathbb{R}^2$ and $A = \{(0, 0)\}$, then $H_1(X - A; G) \cong G$ because $X - A$ is homotopic to a circle, but $H_1(X, A; G) \cong \tilde{H}_1(X; G) = 0$.

Is the above statement true in this generality? If so, where can I find a proof, and where is my attempt at a counterexample mistaken? If not, what's the correct formulation?

(The main case I'm interested in is where $X$ is a complex projective variety and $A$ is a closed subvariety containing the singular locus of $X$.)

$\endgroup$
  • $\begingroup$ I woke up in the middle of the night randomly and I'm too tired to actually think but to apply Poincare duality in the form you did you need a compact manifold, don't you? Have you looked in Hatcher? I believe he has a statement about Lefschetz duality but I'm not sure it's the same as this one. I am also interested in an answer to your question by the way. $\endgroup$ – Seth Apr 14 '14 at 7:37
  • $\begingroup$ The website says that the statement is in S. Lefschetz, "Manifolds with a boundary and their transformations" Trans. Amer. Math. Soc. , 29 (1927) pp. 429–462. I'm not sure how easy it would be to get your hands on that. $\endgroup$ – Seth Apr 14 '14 at 7:40
  • $\begingroup$ @Seth: Poincaré duality works fine here; to use Poincaré duality for a non-compact oriented manifold, you just need to replace cohomology with compactly supported cohomology. (This is Theorem 3.35 in Hatcher.) As for the paper of Lefschetz, I have a copy of that, but it only seems to deal with the case of manifolds with boundary, so I don't see how to extract the statement I described from it (though I could have missed something — it's written in fairly old-fashioned language). $\endgroup$ – Daniel Hast Apr 14 '14 at 7:45
  • 2
    $\begingroup$ FYI the article in question is available freely here. $\endgroup$ – Najib Idrissi Apr 14 '14 at 13:40
  • $\begingroup$ I guess that version of Poincare duality follows from Lefschetz duality as stated. I can't find anything wrong with what you said so I'm confused. $\endgroup$ – Seth Apr 14 '14 at 14:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.