Understanding Poincare Duality

Question

Suppose $M$ is a oriented closed manifold (in fact to make things concrete, let's just fix the dimension as 3). I have read a proof of Poincare duality but I do not think I have developed any (geometric) understanding for what it says. I tried to read the "dual complexes" proof sketch on wikipedia (under poincare duality) but it did not make sense to me.

I would be very happy if someone would tell me what pictures they have in there head when thinking of Poincare duality. For example, given an explicit 2-cycle $[\alpha] \in H_2(M; \mathbb{Z})$ and an explicit 1-chain $a \in C_1(M; \mathbb{Z})$, what is $PD([\alpha])(a)$ geometrically represent?

I currently operate under the philosophy of "Thinking Simplicially and Proving Singularly (and I guess computing cellularly)" so I am more than happy with any explanations in terms of simplicial homology.

Thanks!

Answer 1

It is an intersection pairing. A triangle and a line segment in $3$-space are dual if they intersect in one point. Likewise two line segments in $2$-space are dual if the intersect is a point. In both cases a single point is dual to the $2$ or $3$ dimensional simplex that contains it (usually as barycenter).

If you think of it as defined by the cap product with the cycle $[M]$, then to an simplex in say $n$ dimensional space you take its geometric complement, meaning for example if you have a triangle in space its complement is the line segment passing through the interior of the triangle, and conversely.

This kind of duality in euclidean geometry where a $k$ dimensional subspace is paired with an $n-k$ space such that together the span $n$ space (orthogonal, if you want) was known before Poincare, and I believe it was one of his motivations.