Let $K$ be Klein bottle, $T$ a torus with one hole and $F$ a surface which contains Möbius strip (as a subspace). Show that $F\#T\cong F\#K$ (and in fact if $P$ is the projective plane $P\#T\cong P\#P\#P$).
In the lecture we mentioned that $K=P\#P$ since is a disc is removed from the projective plane, we get a Möbius strip. so we need to prove that $$F\#T\cong F\# P\#P$$. That's enough to define the isomorphism $\varphi:F\#T\to F\#K$ on the generators. Suppose $$F\#T=\langle y_1,y_2,x_3,\dots x_n\mid y_1y_2=y_2y_1,R_1,R_2,\dots R_s \rangle$$ I think that intuitively $\varphi(x_k)=x_k$ so it will satisfy the relations in the $F\# K$ but I with the other generators and relations I'm stuck.
How can I depict myself the two spaces (How they look like exactly?) and how I define the isomorphism between them?
One good way is to represent surfaces as quotients of squares with opposite sides properly identified: torus $T$ respecting orientation, projective plane $P$ reversing, $K$ Klein bottle one respecting another reversing. Then you cut in each of a disc touching the sides in exactly one vertex, and it remains to glue $P$ ro each of the others by the border of the missing disc. The point here is to cut the drawings even more, reflect or move the pieces and glue them again keeping track carefully of the identifications. In the end the representations of $P\# T$ and $P\# K$ will coincide. This is difficult the say in words... I upload this very drafty drawing: