Quantitative Smoothing of Polyhedral Manifolds

We use a recent result of C.

Lange to obtain a converse to a theorem of B.

Bowditch in dimension at most $4$.

In particular, we show that, for $n \leq 4$, a polyhedral $n$-manifold $X$ with bounded geometry is $K$-bi-Lipschitz homeomorphic to a Riemannian manifold $M$.

We bound the constant $K$, the curvature, and the injectivity radius of $M$ by the bounds on the geometry of $X$.