The problem I face is proving homotopy equivalence between $S^n/S^k$ and $S^n\vee S^{k+1}$.
I've made a proof, but I don't know if it is correct or even if it is a proof at all (perhaps there's too little math in it).
My attempt was to define $Y=S^n\cup D^{k+1}$. Then $Y/D^{k+1}$ should have the same homotopy type as $S^n/S^k$ because they would be both "made" the same way.
Then, since $D^{k+1}$ is a contractible simplicial subcomplex in $Y$, $Y$ is equivalent (homotopy) to $Y/D^{k+1}$. Now if we set $P$ to be a semisphere over $D^{k+1}$, as $D^{k+1}$ is contractible in $Y$ so is $P$.
But $Y/P\simeq S^n\vee S^{k+1}$ because the disc's boundary shrinks to a point (and $D^{k+1}/P\simeq S^{k+1}$).
What do you think? Is it conceptually fine? Moreover is it mathematically sufficient and correct? I've always found problems in topology difficult in that aspect.