triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant) 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.
 A: 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...
A: 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.
A: The minimal triangulation of a torus has 1 vertex, 3 edges and 2 triangles, if one allows an edge to have equal start and ending point: Draw a square and its diagonal, glue together its corresponding edges to create a torus.
Note that this triangulation is degenerated from the standpoint of simplicial homology since the triangles are not homeomorphic to a standard 2-simplex in affine space (whose vertices are distinct). However, this does not influence the Euler-characteristik, which can be defined for any graph with simply connected faces.
