Analogous of Poincaré Duality for relative homology and relative cohomology

Question

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?

Thank you for your time.

Answer 1

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 $M/A$.

Answer 2

You can find a formulation of relative Poincare Duality for instance in Section 7 (pages 291-298) of

Dold, Albrecht, Lectures on algebraic topology., Classics in Mathematics. Berlin: Springer-Verlag. xi, 377 p. (1995). ZBL0872.55001.

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.