[Math] Lefschetz duality for non-compact relative manifolds

ag.algebraic-geometryat.algebraic-topologyreference-request

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 $X$.)

I asked this question on Math.SE already, but there are still no answers after 10 days.

Also, I'm aware of the similar formulation of Lefschetz duality proved on p. 297 of Spanier's Algebraic Topology, but it's not quite the same, and I don't see how one follows from the other.

Best Answer

Your counterexample looks correct to me. I don't see an obvious way to add hypotheses on $X$ or $A$ so that the statement called "Lefschetz duality" in the link you give becomes correct.

Some correct versions of Poincar\'e(--Lefschetz) duality are the following:

  1. $M$ a closed orientable manifold, then $H^i(M)=H_{n-i}(M)$.
  2. $M$ a compact orientable manifold with boundary, then $H^i(M)=H_{n-i}(M,\partial M)$ and $H^i(M,\partial M)=H_{n-i}(M)$.
  3. $M$ a (not necessarily compact) orientable manifold, then $H^i_c(M)=H_{n-i}(M)$, and $H^i(M)=H_{n-i}^\mathrm{lf}(M)$ ("locally finite" homology, also known as Borel--Moore homology).
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