Cap Product and Poincaré Duality

Orientability

Definition

A manifold $M$

Lemma

Let $M$ be an $n$-dimensional oriented connected compact manifold. Then

• If $M$ is $R$-orientable then $H_n(M;R)\cong R$ and $H_n(M;R)\to H_n(M{|}_x;R)$ is an isomorphism for all $x\in M$.
• If $M$ is not $R$-orientable then $H_n(M;R)\cong R[2]=\{r\in R:2r=0\}$.
• $H_k(M;R)\cong 0$ for all $k>n$.
  

Fundamental Class

Let $M$ be an $n$-dimensional oriented connected compact manifold. The fundamental class of $M$, denoted $[M]$, is the unique generator of the top-dimensional homology group $H_n(M;R)$ that corresponds to the given orientation of $M$.
More concretely, the orientation of $M$ is a choice of an isomorphism $\iota :R\to H_{n(M;R)}$, and $[M]:=\iota (1_R)$.

e.g.

• For a positively oriented point $p^{+}$, the fundamental class is $[p^{+}]=1\in H_0(p;\mathbb{Z})\cong \mathbb{Z}$. If we instead give it the opposite orientation, the fundamental class would be $[p^{-}]=-1$.

Cap Product

Cap Product

For $k\ge l\ge 0$, we have a bilinear map called the cap product: $⌢:C_k(X;R)\times C^l(X;R)\to C_{k-l}(X;R),$ which sends $(\sigma ,\phi )$ to $\sigma ⌢\phi =\phi (\sigma {|}_{[v_0,\dots ,v_l]})\cdot \sigma {|}_{[v_l,\dots ,v_k]}$.
This induces a map on homology and cohomology: $⌢:H_k(X;R)\times H^l(X;R)\to H_{k-l}(X;R).$

Compatibility with Cup Product

$x⌢(\alpha ⌣\beta )=(x⌢\alpha )⌣\beta$.

Poincaré Duality

If $M$ is an $n$-dimensional oriented compact manifold, then the $k$th cohomology group of $M$ is isomorphic to the $(n-k)$th homology group of $M$: $H^k(M)\cong H_{n-k}(M).$
Specifically, this isomorphism is given by the cap product with the fundamental class $[M]\in H_n(M)$: $\phi \mapsto \phi ⌢[M],$ where $\phi \in H^k(M)$ and $\phi ⌢[M]\in H_{n-k}(M)$.