triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant) - MathOverflow

triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant)

2010-04-25T03:52:32Z

Hi, everyone:

I have been going over some simplicial homology recently, hoping to get
some geometric insight that I don't know how to get from the algebraic machinery alone.

I have been trying to find the homology of the torus this way, i.e., by triangulating it ( i.e., finding a carrier for the torus), but the smallest triangulation I have been able to do , has 18 triangles/faces --I checked it works; there are 8 vertices and 26 edges. Still: does anyone know of a simpler triangulation, ie., one with a smaller total number of triangles (and, of course, fewer vertices and edges resp.). ?

I had tried the long shot of solving the very simple equation:

V-E+F =0

in positive integers.

but this alone does not seem to help . Any ideas.?. Any ideas for finding minimal triangulations of surfaces, or higher-dimensional manifolds.?

Thanks.

Answer by David Eppstein for triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant)

2010-04-25T04:12:08Z

If you're just looking to glue triangles together along their edges, you can do it with two triangles, glued together to form a square, and then with opposite sides of the square glued to form a torus in the usual way. The resulting mesh has one vertex and three edges.

But if the triangles have to form a simplicial complex (meaning that the intersection of any two triangles is empty, a single vertex, or an edge) then I think the smallest mesh for a torus has 14 triangles, connected to each other in the pattern of the Heawood graph. The resulting mesh has seven vertices and 21 edges. It can be embedded into space as the Császár polyhedron.

Answer by John Palmieri for triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant)

2010-04-26T17:28:48Z

For the particular case of a simplicial complex structure for a torus, David Eppstein is right: the minimal triangulation has 7 vertices, 21 edges, and 14 triangles. For a sphere, the minimal triangulation has $(v,e,f) = (4, 6, 4)$. For a real projective plane, the minimal triangulation has $(v,e,f) = (6, 15, 10)$.

For the general situation of finding minimal triangulations of manifolds, Frank Lutz has written a nice preprint, and he also has some information and other references on The Manifold Page. There are plenty of unsolved problems in this area, it seems...