= Connected sum of oriented manifolds
{title2=$M\mathbin\#N$}
For connected oriented closed $d$-dimensional <manifolds> with $d\geq2$, their connected sum removes an open $d$-dimensional ball from each and identifies the resulting boundary <spheres> by an <orientation-reversing diffeomorphism>. The remaining orientations fit together. Collapsing the separating sphere gives a <pinch map> to the <wedge sum> of the two closed manifolds, and projection to either summand has <degree of a continuous mapping> one. In intermediate positive degrees the <cohomology> is the direct sum of the summand groups. Products of classes from different summands vanish; products landing in top degree use the single common <orientation class>. These statements follow from <excision>, the <Mayer–Vietoris sequence> and the degree-one projections.
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