My question is really basic. I was reading on the Massey's Book of algebraic topology on the connected sum of the torus and real projective plane, I felt a little confused his explanation. Someone can explain to me Or at least give a reference where I can find this calculation?
Since I don't have Massey's book, I'm not sure what he does, and you haven't given any clue as to what it is that confuses you. But there is a proof that the connected sum of a torus and a projective plane is homeomorphic to the connected sum of three projective planes on page 9 of these notes. All the definitions are given earlier in that chapter.