Poincaré duality II (Ex)
From Manifold Atlas
For a simplicial complex , define the front -face of an -simplex as and the back -face as
The Alexander-Whitney diagonal approximation is given by
and the partial evaluation map is defined as
Recall, we define the cap product on the chain level by
and this descends to a well defined product on (co)homology. Consider as a simplicial complex with three -simplices and three -simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map
thus verifying that has Poincaré duality.