[GAP Forum] van Kampen diagrams (fwd)

[Petra Holmes]

Just to say that our question is answered by this nice counterexample from 
David Hobby:

---------- Forwarded message ----------
Date: Mon, 08 Jan 2007 14:12:43 -0500
From: David Hobby <hobbyd at newpaltz.edu>
To: Petra Holmes <holmespe at for.mat.bham.ac.uk>
Subject: Re: [GAP Forum] van Kampen diagrams

Petra--

May we have a precise definition of "polyhedral ball", just
for clarification?  I've got a counterexample, but it's
not what most people would call a polyhedron.  It is, however,
derived in an obvious way from a planar graph, which is what
van Kampen diagrams seem to be.

Is it legal for two faces to touch at more than one shared
edge?  If so, my counterexample consists of something that
can be visualized as the globe, with a 15-gon covering almost
all the Northern hemisphere, another one covering almost
all the Southern hemisphere, and a ring of 5 "diamonds" going
along the equator.  Each diamond looks like this, where up is
North:

             .
            /|\
_________./ | \._______________
           \ | /
            \|/
             .

(This mess can only be deciphered if you have the spacing set
the right way in ASCII mode.  Sorry!)

Then the two 15-gons touch at 5 separate edges, running
along the equator between the 5 diamonds.  (Each diamond is
made of two triangles which share a north-south edge, in
case my diagram didn't work.)

Nice problem.

 					---David