[GAP Forum] van Kampen diagrams

[Dmitrii Pasechnik]

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" <holmespe at for.mat.bham.ac.uk> 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 <RParker at amadeuscapital.com>
> To: Beth Holmes <P.E.Holmes at dpmms.cam.ac.uk>
> 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.