I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism.
It's clear that an element of $\pi_r(SO(q)$ gives rise to a map $S^r\times S^{q-1} \rightarrow S^{q-1}$. After doing a suspension, this becomes a map $S(S^r\times S^{q-1})\rightarrow S^q$. However I have trouble understanding or visualizing the next step, that is $S(S^r\times S^{q-1})\simeq S^{r+q}/S^r\times S^{q-1}$.
For example, in the case $r=1,q=2$, this is saying $S(S^1\times S^1)\simeq S^3/T^2$. Since $S^3$ can be realized as two solid tori, I think this means that, identifying the $T^2$ which is the common boundary of the two solid tori, gives rise to $S(S^1\times S^1)\sim S^2\vee S^2\vee S^3$. Again I don't know how to visualize this.
Any help is appreciated. Thanks.