----
$R$ is a ([[commutative ring|say, commutative]]) [[ring]].
> [!theorem] Theorem. ([[Poincare duality]])
> Let $M$ be an $R$-[[(homological) orientation of a manifold|oriented]] $d$-[[manifold]]. There is a canonical **duality map** $D=D_{M}: H^{k}_{c}(M;R) \to H_{d-k}(M; R)$
and this map is an [[isomorphism]].
>
If $M$ is [[compact]], $H^{k}_{c}(M;R)=H^{k}(M;R)$ and there is a [[The Thom Theorem for oriented manifolds|fundamental class]] $[M]$, then $D_{M}$ is easily defined via the [[cap product]]: $D_{M}([\varphi]):= [M] \frown [\varphi].$
In general, one defines for each [[compact]] [[subspace topology|subspace]] $K \subset M$ a map $D_{M}^{K}: H^{k}(M | K; R) \to H_{d-k}(M ; R)$ as $\mu_{K} \frown -$[^1], then glues these maps into $D$. Details below.
^theorem
[^1]: Recall that the (2nd) [[relative cap product]] turns two [[relative singular homology|relative classes]] into an [[singular homology|absolute class]].
> [!proposition] Corollary.
> If the $d$-manifold $M$ is moreover [[compact]]: $H^{k}(M; R) \cong H_{d-k}(M;R).$
^corollary
> [!definition] Definition of the duality map $D_{M}$.
> We suppress all the '$;R