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?
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$\begingroup$ Yes, this is a theorem of Whitney's. $\endgroup$– Ryan BudneyCommented Mar 24, 2016 at 3:41
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$\begingroup$ Do you have a reference that might be useful? $\endgroup$– BlakeCommented Mar 24, 2016 at 3:43
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$\begingroup$ Plenty. mathoverflow.net/questions/8789/… $\endgroup$– Ryan BudneyCommented Mar 24, 2016 at 3:45
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$\begingroup$ 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. $\endgroup$– BlakeCommented Mar 24, 2016 at 3:49
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$\begingroup$ 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. $\endgroup$– Ryan BudneyCommented Mar 25, 2016 at 0:33
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