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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!

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    $\begingroup$ It morally represents "intersection number with the 2-cycle". $\endgroup$ – user98602 Sep 8 '16 at 21:39
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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.

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    $\begingroup$ Yes, the intersection pairing was the motivation for the cup product. This probably was even the reason for introducing cohomology (homology was defined much earlier). This I think is discussed in "A History of Algebraic and Differential Topology" by Dieudonné. $\endgroup$ – Moishe Kohan Sep 10 '16 at 14:49

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