homotopy theory - geometry of null homotopies

Your question should follow from: given a homotopically trivial map $g:S^nto S^m$, is there a homotopy $h: D^{n+1} to S^m, h_{| partial D^{n+1}} = g$ such that $textrm{stretch}(h) leq textrm{stretch}(g)$? Or maybe there exists a reasonable function $L_n:[0,infty) to [0,infty)$ such that $textrm{stretch}(h) leq L_n(textrm{stretch}(g))$. I think you're claiming that there exists $L_n$ which is computable. It seems to me that since $textrm{stretch}(f^k)leq kcdottextrm{stretch}(f)$, the fact that $g=f^k$ doesn't really matter in the problem.

Interpreting the problem this way, one approach is to try to find for $zin S^m$ a factorization $g': S^n to T_z S^m cong mathbb{R}^m$, such that $g=exp_zcirc g'$ using the fact that $g$ is homotopically trivial. If one could control the diameter of the image of $g'$, then one ought to be able to bound the stretch of the trivial filling by coning to the origin in $T_z S^m$ (this amounts to filling in $g$ by coning by geodesics to a point). I don't know when such a lift is possible though, since obviously $exp_z: mathbb{R}^m to S^m$ is far from being a fibration at concentric spheres about the origin.

Here's a possible approach: Take a map $h: D^{n+1} to S^m, h_{| partial D^{n+1}} = g$ where $h(0)=p$. Think of this a map $H: S^{n} to [([0,1],0), (S^m,p)]$, that is a map from $S^{n}$ to the space of intervals at p. One may show that we may homotope $h$, keeping $h_{| S^{n}=partial D^{n+1}}$ fixed, to a map sending each interval $[0,1]x, xin S^{n}$ to a piecewise linear path. This is the process described in Theorem 17.1 of Milnor's book. The idea then would be to try to homotope all of the intervals to be piecewise geodesics of bounded length (above I suggested homotoping all of the intervals to geodesics, which is absurd), and then try to show that the resulting map has bounded stretch. Like the fundamental theorem of Morse theory, I would expect to be able to homotope things down onto $n+1$-cells which correspond to index $n+1$ geodesics, and therefore have bounded length, by something like cellular approximation. But I don't quite know how to complete the argument yet.

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