Triangulation of Torus, Algorithms for Calculating Simplicial Homology

[Bacle]

Hi, everyone:

A couple of points, please:

1) I am reviewing last semester's Simplicial Homology. I was able to do a triangulation
of the torus T²=S¹xS¹ , and I was able to do
a triangulation of T² , 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.

 

[jvicens]

Dear forum members,

Thank you for sharing your thoughts and questions on Simplicial Homology. I am glad to see that you are actively exploring and reviewing this topic. I would like to address your points and offer some guidance and suggestions.

1) Triangulation of the Torus: It is great that you were able to successfully triangulate the torus with 18 triangles. However, I understand your desire to find a smaller triangulation. While I am not aware of a triangulation with fewer than 18 triangles, I suggest looking into the work of mathematician Branko Grünbaum, who studied minimal triangulations of surfaces. His paper "Minimal triangulations of closed surfaces" (1970) may be of interest to you. As for finding the Euler number of general surfaces, there are a few approaches you can take without using homology. One method is to use the Gauss-Bonnet theorem, which relates the Euler number to the Gaussian curvature of the surface. Another approach is through the use of the Euler characteristic, which can be calculated using the number of vertices, edges, and faces of a triangulation. I suggest exploring these methods further and see if they can help you find the Euler number of the Mobius band without relying on homology.

2) Algorithm for Simplicial Homology: Yes, there are algorithms to compute the Simplicial Homology of a space X, once it has been triangulated and the chain groups are known. One approach is through the use of matrices and linear algebra, as you mentioned. Another method is through the use of computer software, such as the software package "Simplicial Homology" in Mathematica. I suggest looking into these methods and see which one works best for you.

Overall, it is important to keep in mind that triangulation and homology are tools that complement each other in studying topological spaces. While triangulation helps us understand the geometric structure of a space, homology allows us to capture its topological properties. I hope this helps and good luck with your studies. Feel free to reach out if you have any further questions or need more guidance. Happy exploring!