Connected Sums

The projective plane with two handles can be formed as the connected sum of
a projective plane with two tori, $M = P\+T\+T$.
But $P\+T\cong P\+K$, where $K$ is a Klein bottle, so $M=P\+K\+T$. We want to make a connected sum of a projective plane and a Klein bottle with a handle. |