Subject: Re: flat triangulation of torus
Date: Mon, 12 Feb 2001 15:42:06 +0530 (IST)
From: "Prof. A. R. Shastri" <ars@math.iitb.ernet.in>
CC: joellis@emba.uvm.edu

Some thing closely related to this has been up in mind for past few
months. First of all, thank you for the information on  Scaszar's work. I
would like to know the exact reference to this material.

In December 2000, at Allahabad conference on Low dimensional Topoogy, I
gave a talk on `why PL?' In this talk I actually displayed PL models of a
torus and a genus 2 orientable closed surface. I also showed how  one
can easily extend my constructions to any genus. Moreover, I also showed
 how to make PL  models of non orientable surafces with boundary
component. From this I think you can conclude that evry surface that can
be embedded in R^3 can be (flatly)  PL-embedded. My triangulated model of
the torus has 9 vertices and I asked the audiance whether any of them know
about an embedding of K(7) or some other PL embedding of torus with eight
vertices. It seems nobody was aware of Scaszars' work.

I have not written down this stuff even though some participants
explicitly asked me to do that thinking that experts in this area may
already know such things.

Anant R Shastri

 On Thu, 8 Feb 2001, Donald Davis wrote:

> Subject: question (fwd)
> Date: Thu, 8 Feb 2001 13:42:47 -0500 (EST)
> From: Joanna Ellis-Monaghan <joellis@emba.uvm.edu>
> CC: Dan Archdeacon <archdeac@emba.uvm.edu>
>
>
> Joanna A. Ellis-Monaghan
>
> e-mail:  joellis@emba.uvm.edu or jellis-monaghan@smcvt.edu
>
> website: http://academics.smcvt.edu/jellis-monaghan
>
> Department of Mathematics        phone:  802 656 2940
> University of Vermont              fax:  802 656 2552
> 16 Colchester Avenue
> Burlington, VT  05405
>
> or
>
> Department of Mathematics          phone:  802 654 2660
> Saint Michael's College                 fax:  802 654 2610
> Winooski Park
> Colchester, VT  05439
>
> _____________________________
> Torus flat triangulation question
>
> We are looking at the following problem about polyhedra in Euclidean
> 3-space.  Let us first describe an example.  Consider the well-known
> triangulation of a torus by the complete graph K(7).  This has 7
> vertices,
> 21 edges, and 14 triangles.  In the 50's Csaszar found what we call a
> _flat realization_ of this polyhedron in 3-space.  That is, he located
> the
> vertices in 3-space and represented each face by the convex hull of its
> three boundary vertices, so that the intersection of any two faces in
> 3-space was exactly the convex hull of the intersection of their
> boundary
> vertices.  Hence each face is contained in a plane, and two faces have
> no
> interior intersection.
>
> We would like to show that any simple triangulation of the torus has
> such
> a flat realization.  This is in the flavor of Steinitz' Theorem, which
> says that a graph is the 1-skeleton of a (spherical) convex polyhedron
> in
> 3-space if and only if the graph is 3-connected and planar. Toroidal
> polyhedral, of course, would not be convex.  We are focusing on
> triangulations as the most interesting (easiest?) special case.
>
> Does anyone know of work done in this area?  We would be particularly
> interested in a proof that there exist *any* polyhedron at all
> homeomorphic to an orientable surface, regardless of the genus, that can
>
> not be realized in 3-space by flat faces.  What happens if the faces are
>
> not constrained to be triangles?
>
> We would be interested in any results along these lines.  If you know
> anything, please contact dan.archdeacon@uvm.edu or joellis@emba.uvm.edu.
>
> Thanks for your attention to our question.
>
> -------------------------------------------------------------
>   Dan Archdeacon                        Tel:  x6-0850 (work)
>   204 Mansfield (office)                     872-0023 (home)
>   Dept. of Math. & Stat. (campus mail)        x6-0696 (fax)
>   dan.archdeacon@uvm.edu  http://www.emba.uvm.edu/~archdeac/
> -------------------------------------------------------------
>
>