
Topology
Why is it obvious that attaching an (m+n)cell to SmVSn is SmxSn?
Also, if you have a space X that is the real line, with two sphere's (S2) wedged at each integer, to determine its homology group, I figured I should use the Mayer–Vietoris sequence, so I split it such that U = X  {odd integers}, V = X  {even integers}. Do you think that's what I should do?

Re: Topology
For your first question, note that , that is, is the one point compactification of .
Now
(disjoint union)
First glue we get , then glue by identifying it with the unit disk, then glue it to the frame. It's not easy to express but you can always take m=n=1 as an example to study the procedure to glue (1 square)+(2 segments)+(1 point) together to get a torus.