# Are characteristic maps of CW complexes Lipschitz up to homotopy?

Let us consider a finite CW complex $X=\cup X_j$ with a given metric, compatible with the topology (maybe a reasonable one coming from some embedding into some $\mathbb{R}^n$). The characteristic maps are the maps $\phi_j^k:\mathbb{S}^{j-1}\to X_{j-1}$ for $k=1,\ldots, N_j$ which prescribe how to glue the boundaries of the $j$-dimensional cells $e_j^1,\ldots,e_j^{N_j}$ to the $(j-1)$-dimensional skeleton of $X$.

Do there always exist maps $\psi_j^k:\mathbb{S}^{j-1}\to X_{j-1}$ which are homotopic to $\phi_j^k$s and Lipschitz? If the answer is negative, in general, under which hypothesis on $X$ could this be true?

NOTES

1. If $X$ is actually a simplicial complex, by simplicial approximation this should be true.
2. If $X$ is a CAT(k) space, this is probably true by baricentric subdivision.

Let $X$ be the mapping cone of a space-filling curve $S^1\to D^2$ (here $D^2$ is the 2-dimenionsal disk). Then $X$ is homotopy equivalent to $S^2$, but I doubt that you can homotope the map $S^2\to X$ to make it Lipschitz.
• Which metric do you chooose on $X$ ? – HenrikRüping Sep 11 '13 at 14:10
• I was thinking of the metric coming from some embedding in $\mathbb R^n$ (as suggested in the question). – André Henriques Sep 11 '13 at 15:41