I have two homotopies $H,G:D^n \times I \to Z$ with $H(x,0) = f(x)$, $H(x,1) = f'(x)$, $G(x,0) = f'(x)$, $G(x,1) = g(x)$ for some maps $f,f',g:D^n \to Z$.
$G$ has the additionnal property of being constant (in $t$) on $S^{n-1}$, i.e $$ G|_{S^{n-1} \times I} = f' |_{S^{n-1}} = g|_{S^{n-1}}. $$
Is it possible to combine them in a homotopy $H'$ from $f$ to $g$ in such a way that $H'$ actually coincides with $H$ on $S^{n-1}$ ? I.e such that $$ H' |_{S^{n-1}\times I} = H |_{S^{n-1} \times I}. $$
I know how to combine them to get a homotopy from $f$ to $g$, and this homotopy will then do the exact same thing as $H$ on $S^{n-1}$, the only problem is that it will do it twice as fast. So I thought of "slowing down" the homotopy as $||x||$ approaches 1 (i.e as $x$ approaches $S^{n-1}$) but I do not know if that is a good idea...
Thanks for your help!
Best, Olivier