Can you provide me with any intuition behind the Cap product of a cohomology class and a homology class? What is its geometric meaning? Can you also give me an intuition why the Poincaré duality is true?
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I am not sure this is the geometric meaning you want to have, but it's very helpful for me to understand Poincare duality: I think that the baby version of Poincare duality is the duality for planar graph (or a graph embedded in a general $2$-dimensional surface). Given a planar drawing of $G$, then the dual graph $G^*$ is a graph which has a vertex for each plane region of $G$, and an edge for each edge in $G$ joining two neighboring regions, according to Dual graph. Of course, the plane is $2$-dimension. Therefore, face in $G$, which is $2$-dimensional, corresponds to a vertex in $G*$, which is $2-2=0$-dimensional. Edge in $G$, which is $1$-dimensional, corresponds to an edge in $G*$, which is $2-1=1$-dimensional. And vertex in $G$, which is $0$-dimensional, corresponds to a face in $G*$, which is $2-0=2$-dimensional. On the other hand, $|V(G)|=|F(G^*)|$, $|E(G)|=|E(G^*)|$, and $|V(G^*)|=|F(G)|$, which is Poincare duality.