If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:05:21Z http://mathoverflow.net/feeds/question/92286 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92286/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? Ernest Davis 2012-03-26T16:11:08Z 2012-03-26T18:55:24Z <p>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 <em>The Hauptvermutung Book</em>. 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. </p> <p>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. </p> http://mathoverflow.net/questions/92286/if-a-polyhedron-is-homeomorphic-to-a-simplex-is-it-piecewise-linear-homeomorphic/92289#92289 Answer by Igor Rivin for If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic? Igor Rivin 2012-03-26T16:32:58Z 2012-03-26T16:32:58Z <p>I am not sure if this is what you are asking, but check out <a href="http://en.wikipedia.org/wiki/Exotic_R4" rel="nofollow">http://en.wikipedia.org/wiki/Exotic_R4</a> (note that in dimension four, PL is the same as smooth).</p> http://mathoverflow.net/questions/92286/if-a-polyhedron-is-homeomorphic-to-a-simplex-is-it-piecewise-linear-homeomorphic/92297#92297 Answer by Dmitri for If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic? Dmitri 2012-03-26T18:55:24Z 2012-03-26T18:55:24Z <p>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. </p> <p>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. </p>