If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic? - MathOverflow most recent 30 from http://mathoverflow.net2013-06-19T06:05:21Zhttp://mathoverflow.net/feeds/question/92286http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/92286/if-a-polyhedron-is-homeomorphic-to-a-simplex-is-it-piecewise-linear-homeomorphicIf a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?Ernest Davis2012-03-26T16:11:08Z2012-03-26T18:55:24Z
<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#92289Answer by Igor Rivin for If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?Igor Rivin2012-03-26T16:32:58Z2012-03-26T16:32:58Z<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#92297Answer by Dmitri for If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?Dmitri2012-03-26T18:55:24Z2012-03-26T18:55:24Z<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>