Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Have you looked at the Wikipedia page? It describes the cap product as a type of partial evaluation function. Regarding why Poincare duality is true, see:… , which you can interpret in terms of cap products if you like. – Ryan Budney Dec 5 '11 at 6:14
up vote 3 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.