Understanding Poincare Duality

Solution 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.