# If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?

If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic? In particular, is this true in $R^{4}$? In 2 and 3 dimensions any two polyhedra that are homeomorphic are PL-homeomorphic, by theorems of Rado and Moise. In dimension $\geq 5$, this is a trivial special case of theorem 1.1 in M.A. Armstrong "The Hauptvermutung According to Lashof and Rothenberg" in The Hauptvermutung Book. But I have not found a statement that covers it for dimension 4; and I am not confident that dimension 4 can easily be reduced to dimension 5.

Also, if anyone can suggest a reference for this particular case that does not go through these very high-powered, difficult, general theorems, I would be interested on stylistic grounds.

-

If you assume that your polyhedron has only finite number of faces, I think the answer to your question is unknown. Moreover any answer to such question would give a solution to Smooth Poincare conjecture in dimension 4, which is still open.

Indeed, suppose you have a four-dimensional sphere with an exotic smooth structure. Then you can always triangulate such a sphere in a finite number of simplexes. Now, throw away a simplex from such a triangulation. What you get is a homeomorphic to a simplex, but can not be PL diffeomorphic to it, otherwise your initial sphere would be PL diffeomerphic to the standard one, which is not sow since you sphere is exotic.

-
But the original question isn't about PL diffeomorphic, it's about PL homeomorphic. Are these the same concept in dimension 4? For example, this argument would not work in dimension 7 where a triangulation of an exotic 7-sphere would have to be PL-homeomorphic to the standard sphere by the PL Poincare conjecture. (For that matter, what does "PL diffeomorphic" mean exactly?) –  Greg Friedman Mar 26 '12 at 22:06
Gerg, I guess we just use two different terms to define the same notion. I agree with you about dimension 7, there is only one PL structure on $S^7$. But dimension $\ge 7$ are different from $<7$, namely, up to dimension 6 every PL manifold admits a unique smooth structure. In particular all these exotic smooth 4-dimensional manifolds have have exotic PL structures. –  Dmitri Mar 26 '12 at 23:04