# 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.