Pinch map ($M\#N\to M\vee N$)

A pinch map collapses a separating subspace to a point so that the quotient splits into a wedge sum. For a connected sum of oriented manifolds, collapsing the separating boundary sphere gives a map $M\#N\to M\vee N$. The two wedge projections have degree of a continuous mapping one with compatible orientations. Consequently their pullbacks identify the two top orientation classes, while positive-degree classes from different summands have zero cup product.