I've encountered the following problem that I'm sure someone more topologically inclined can answer:
Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ is good (I don't know the exact term for it if there is one) if the homotopy is constant in time outside of some coordinate chart in $Y$, in which case we'd say the homotopy is supported in the chart.
Suppose you have two continuous maps $f_0$ and $f_1$ between compact smooth manifolds $X$ and $Y$ which are homotopic. Cover $Y$ with finitely many coordinate charts $U_\alpha$. My question is the following:
Are $f_0$ and $f_1$ homotopic with respect to a finite sequence of good homotopies supported in the elements $U_\alpha$ of the covering? Can the number of good homotopies required be uniformly bounded by topological data like the homotopy class of $f_0$ or the number of elements in the covering?
I am interested in this question because I have a functional which I can get a good estimate for if the above is true.