Let $M$ be a simply connected complete Riemannian manifold, and let $x\in M$. Does there exist a nondecreasing function $R:\mathbb R\_+\to\mathbb R\_+$ such that, for every $r>0$ and all paths $\alpha,\beta$ in the ball $B(x,r)$ from $x$ to any $y\in B(x,r)$, there is a homotopy relative to the end points between $\alpha$ and $\beta$ in the ball $B(x,R(r))$?



Homotopy bounds in simply connected complete Riemannian manifolds
It will follow if we can find a function $R$ so that the map induced by inclusion $\pi\_1 (B(x,r)) \to \pi\_1 (B(x,R(r)))$ is trivial.


By the density of Morse functions, for any small $\epsilon$ there is a compact smooth manifold with boundary $N$ so that $B(x,r) \subset N \subset B(x, r+ \epsilon)$. The fundamental group $\pi\_1 (N)$ is then finitely presented. Now choose $R$ large enough so that each generator of $\pi\_1 (N)$ is nullhomotopic in $B(x,R(r))$. This finite value $R$ is then sufficient for the given $r$.


Now for every $r$ choose $R$ minimal. (It might be that actually choosing the infinum causes some problems, but we could simply add a fixed value $\delta > 0$ to cushion it.) This will guarantee the function is non-decreasing. 


