# Analogous of Poincaré Duality for relative homology and relative cohomology

I am studying Morse Theory on finite dimensional and compact manifolds using homology groups and relative homology groups on $$\mathbb{Z}$$. I want to show that this theory could be developed using De Rham cohomology and relative cohomology. What I'd like to ask you is if it's true that $$\forall k \in \{0,...,dim (\mathcal{M}) \}$$ and $$\forall \, A \subset \mathcal{M}, \, H_k(\mathcal{M},A) \cong H^k (\mathcal{M},A)$$. Regarding the same relation between 'absolute' homology and cohomology, I know the Poincaré Duality $$H_k (M) \cong H_{n-k}(M) \cong (H^{n-k}(M))^*$$ and $$(H^{n-k}(M))^* \cong H^{n-k}(M)$$ (dim $$\mathcal{M} < \infty$$) (Is the previous sequence of isomorphisms correct? I might be wrong due to my inexperience in this topic). If this is correct, can I generalize it to Topological couples and consequently to relative homology? How can I prove this?

• No, the actual relative PD is more complex. Commented Oct 24, 2021 at 12:43
• I will add references later on. Commented Oct 24, 2021 at 15:18
• Ok, thank you. I'll wait for your indication. Commented Oct 25, 2021 at 7:47
• @MoisheKohan and what are the references? I also wanna to know about it. Commented Nov 15, 2021 at 1:01

For simplicity, I will assume that a manifold $$M$$ is oriented and $$n$$-dimensional, $$L\subset K$$ are compact subsets of $$M$$. Then the cup-product with the fundamental class of $$M$$ defines isomorphisms $$\check{H}^i(K,L)\cong H_{n-i}(M-L, M-K).$$ (You can apply this, for instance, in the situation when $$L=\emptyset$$.) Here the left hand side is the Chech cohomology. If $$K$$ is reasonably nice, say, an ANR, then this is the same as the singular cohomology. Dold proves various variations on this result, for instance, one in section 7.12: $$\check{H}_c^i(K)\cong H_{n-i}(M, M-K),$$ where $$K$$ is merely a closed subset of $$M$$. The subscript $$c$$ refers to the cohomology with compact support. Take a look at other versions proven in the same section.
Relative Poincare duality is the statement that for a closed, orientable n-manifold $$M$$ and a compact subset $$A$$ such that $$(X,A)$$ satisfies excision, $$H^k(M,A) \cong H_{n-k}(M-A)$$. This follows from noncompact Poincare duality and the observation that $$M-A$$ is a manifold with one point compactification equal to $$M/A$$.
• I don't understand when you say '$M−A$ is a manifold with one point compactification equal to $M/A$'. Do you mean the compactification of Alexandroff? So are you telling me this sentence is false $∀A⊂M$ ,$H_k(M,A)≅H^k(M,A)$ ? Thank you. Commented Oct 25, 2021 at 7:50
• @Dylan The one point compactification of a locally compact space $X$ is the compact space with a distinguished point such that removing the point gets you $X$. Noncompact Poincare duality relates the homology of $M$ with the cohomology of its one point compactification. And yes, your sentence is wrong for multiple reasons. Poincare duality relates $H^k$ and $H_{n-k}$ for one, but even with this correction you can take $A=\{*\}$ and $k=0$. Then the relative (co)homology is reduced (co)homology and $\bar{H}^0(M)=0 \neq \bar{H}_n(M)=\mathbb{Z}$. Commented Oct 25, 2021 at 13:11