
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 $\displaystyle S^n \cong R^n \cup \{\infty\}$, that is, $\displaystyle S^n$ is the one point compactification of $\displaystyle R^n$.
Now $\displaystyle S^m \times S^n \cong (R^m \cup \{\infty\}) \times (R^n \cup \{\infty\})$
$\displaystyle = R^{m+n} \cup (R^m \times \{\infty\}) \cup (R^n \times \{\infty\}) \cup (\{\infty\} \times \{\infty\})$
$\displaystyle = R^{m+n} \sqcup R^m \sqcup R^n \sqcup \{\infty\}$(disjoint union)
First glue $\displaystyle R^m \sqcup R^n \sqcup \{\infty\}$ we get $\displaystyle S^m \vee S^n$, then glue $\displaystyle R^{m+n}$ by identifying it with $\displaystyle D^{m+n}$ the unit disk, then glue it to the $\displaystyle S^m \vee S^n$ 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.