Non-PL 5-sphere

Description

A PL sphere is a manifold which is PL homeomorphic to the boundary of a simplex. It is known that all the spheres in dimensions less than or equal to 3 are PL, but there are non-PL spheres in dimensions larger than or equal to 5. The famous way to get non-PL sphere is using Edwards' "double suspension theorem", that is, if we take a suspension two times over a homology sphere, we have a manifold homeomorphic to a sphere, but it is not a PL sphere if the original homology sphere is not a sphere.

The data given here is made from Björner and Lutz's triangulation of the poincaré sphere by taking double suspension, according to their paper. They suggested a way to take a suspension by introducing only one new vertex in one time ("Datta's trick"), so this non-PL sphere has only 18 vertices. (With 269 facets.)

Properties

This sphere is not constructible because all constructible spheres are PL. This is Cohen-Macaulay because all spheres are Cohen-Macaulay.

Datum

nonpl_sphere.dat

Some table

vertex decomposable?	no
extendably shellable?	no
shellable?	no
constructible?	no
Cohen-Macaulay?	yes
topology	non-PL 5-sphere
f-vector	(1,18,141,515,930,807,269)
h-vector	(1,12,66,111,66,12,1)
made by	Edwards, triangulated by Björner & Lutz

References

A.Bjorner and F.H.Lutz, Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere", to appear in Experimental Mathematics.

F.H.Lutz, Triangulated manifolds with few vertices and vertex-transitive group actions , Shaker Verlag (1999).