# Smoothing a piecewise smooth manifold

Let $M \subset \mathbb{R}^d$ be a piecewise smooth $2$-manifold. Let $C$ be a polyhedral complex that covers $\mathbb{R}^d$ and contains faces of dimension $[0,d]$. Since $M$ is a $2$-manifold, we can perturb $M$ so that it only intersects faces of $C$ of dimension $d-2$ or greater. Can we smoothen $M$ into a smooth $2$-manifold $\tilde{M}$ such that $d(M,\tilde{M}) \leq \epsilon$ and $\tilde{M}$ intersects every face of $C$ that is intersected by $M$ and no others?

• Yes, this is a theorem of Whitney's. – Ryan Budney Mar 24 '16 at 3:41
• Do you have a reference that might be useful? – Blake Mar 24 '16 at 3:43
• – Ryan Budney Mar 24 '16 at 3:45
• I believe that M could be smoothed, which seems to be what the result you linked proves, but I don't see how this answer my question about the polyhedral complex. – Blake Mar 24 '16 at 3:49
• I think generally you will have to create new intersections when you smooth your map. Think of examples where your original map (of $M$) are highly not-transverse on the skeleta. Even for maps into the plane there appear to be problems. – Ryan Budney Mar 25 '16 at 0:33