This seems to be related to shellability of polytopes, see e.g. G.M.Ziegler's book "Lectures on polytopes" on this topic. (I presume that "topologically a sphere" means that we deal with a 3-connected planar graph. Then by Steinitz thm it comes from a 3-polytope) The "sequence of patches" is then a shelling of the polytope. Shellings of polytopes always exist.
The extra condition on curvature makes the problem seemingly nontrivial. Perhaps some "statistics" on possible distributions of shapes of faces in 3-polytopes might help - but I'm not an expert on this. HTH, -- Dima Pasechnik http://www.ntu.edu.sg/home/dima/ On 1/8/07 6:33 PM, "Petra Holmes" <[EMAIL PROTECTED]> wrote: > Dear Group Pub Forum, > > I have been told to send this to everyone I can think of. Any ideas on it > are welcome. > > Beth > > ---------- Forwarded message ---------- > Date: Mon, 8 Jan 2007 08:26:37 +0000 > From: Richard Parker <[EMAIL PROTECTED]> > To: Beth Holmes <[EMAIL PROTECTED]> > Subject: The central problem > > > Let P be a polyhedral ball which topologically is a sphere, > and where precisely three faces meet at every vertex. > > For each face with n sides, define the "curvature" of that > face to be 6-n, so a pentagon has curvature 1, an octagon > has curvature -2 and so on. The faces do not need to be > regular. > > Euler's formula, F + V = E + 2, implies that the total curvature > over the whole of P is 12. > > A "patch" of f faces is a subset of the faces of P that has > no holes in it. Technically it is simply connected. > > I need to prove or disprove the following result. > > For every such P there is a sequence of patches, each containing > one more face than the previous, such that the sum of the curvature > of all the faces in each patch is greater than zero. In other words > we can build P one face at a time such that the curvature of the > patch we have made so far is always positive. > > This result has huge implications for an algorithm for finitely > presented groups. P is a van Kampen diagram, and I want to know > whether a short relator with a long proof can be built up relator > by relator looking only at sensible things along the way. _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
