- 662

- 1

A couple of points, please:

1) I am reviewing last semester's Simplicial Homology. I was able to do a triangulation

of the torus T

^{2}=S

^{1}xS

^{1}, and I was able to do

a triangulation of T

^{2}, although the best I could do was use 18 triangles.

(the triangulation checked out: 18 triangles/faces, 8 vertices, 10 edges)

I tried to do a triangulation with 12 triangles/faces , 6 vertices, but the edges did not

add up ; I need 18, but could only come up with 15.

Anyone know of a smaller triangulation of the Torus (i.e., with fewer than 18 triangles).?. Anyone know how to find the Euler number of general surfaces (like

the Mobius band) , without using Homology (e.g., Betti numbers.).? . I am

trapped in the catch-22 : to find the (simplicial )Homology, I need to find

a triangulation. But to check if the triangulation is a valid one, I need t find

the Euler number, which I only know how to find using the homology group.

2) Is there an algorithm to find the Simplicial Homology of a space X , once X has

been triangulated, and we know the chain groups of X.?. It seems to come

down to some basic calculations, and some basic linear algebra (row-reduction).

Thanks For any Suggestions/Ideas.