Show that if $X$ is a $m$-connected CW complex and $Y$ is a $n$-connected CW complex then $(X \times Y, X \vee Y)$ is $(m+n+1)$-connected.
I'm trying to use the equivalences in page 346 to state a proof but is being quite hard to set it up so far.. I tried an explicit approach through parametrizations of the sphere but calculations split in too many cases and don't cover all possibilities either... I also thought about using lemma 4.7 but the hypothesis I need to apply it is equivalent to what I'm trying to prove so I'm quite stuck here...
The second part of the exercise is evident through Hurewicz isomorphism theorem so any help on that matter would be highly appreciated. Thanks in advance.
Using your suggestion I came up with this:
Since $X$ is $m$-connected, compression lemma says that the $X^m$ skeleton deformation retracts to any of its $0$-cells since any of their inclusion maps into $X^k$ induce isomorphisms $\forall k \leq m$. This implies $X$ is homotopically equivalent to a CW complex $X'$ with one $0$-cell and all of the others of dimension larger than $(m+1)$. In the same fashion $Y$ is homotopically equivalent to a CW complex $Y'$ with one $0$-cell and all of the others of dimension larger than $(n+1)$.
Noticing that $S^m \wedge S^m \simeq S^{m+n}$ as explained in the definition of smash product, the same argument yields $X \wedge Y \simeq X' \wedge Y'$ with one $0$-cell and the next of dimension larger than $(m+n+2)$ which implies $X \wedge Y$ is $(m+n+1)$-connected.
Finally, since $X \times Y / X \vee Y = X\wedge Y$, apply Hurewicz theorem to obtain isomorphisms $\pi_k(X \times Y, X \vee Y) \rightarrow \pi_k(X\wedge Y)$ if $k \leq m+n+1$ induced by projection to the quotient so $(X \times Y, X \vee Y)$ is $(m+n+1)$-connected. $\Box$
I think it is correct, thank you for the suggestion.