Okay that ideal I wrote about doesn't make any sense. I think I got at least the statement you're claiming.

What do you mean by $(u_M,-u_N)$? Something like $\langle (u_M,0),(0,u_N)\rangle=:I$? And what is the new fundamental class of the connected sum? Something like $(u_M,0)+I=(0,u_N)+I$? How does the cofibration help computing cohomology? But the cofiber isn't actually the wedge but (uncanonically) homeomorphic to it? I'm sorry if these are too basic questions but I wasn't able to figure it out by myself.