i am currently pursuing a course in basic homology theory.i am really stuck in how to triangulate a torus by using simplicial complexes . in every book, a diagram is given but it does not define that why are we subdividing the sheet into a no of rectangles etc,i.e,ther is no explanation given .I know that this is a very basic question but i cannot find a solution on the internet.
if possible,please give an explanation.also,mention references for solved examples on triangulation of a space using simplicial complexes and cw complexes…
thanks a lot
Best Answer
Sorry for the bad quality of the picture. It shows a possible triangulation of the Torus.
Whenever you try to find a triangulation (in particular a Euklidian simplicial complex), it's a collection $K$ of simplicies satisfying the following conditions:
1.) $\sigma\in K$ $\Rightarrow$ every face of $\sigma$ is in $K$.
2.) The intersection of any two simplicies is either empty or a face of both.
3.) For every point in $\sigma\in K$ there exists a neighbourhood which intersects finitely many simplices of $K$.
Let $\{v_0,...,v_n\}$ be the vertices of $\sigma \in K$. A face of $\sigma$ is the simplex spanned by each non-empty subset of $\{v_0,...,v_n\}$.
The most important point is: If you try to triangulate a surface, any two triangles share a single edge or a single vertex or they're disjoint.
A non-example:
This triangulation is false, because the two "triangles" are no triangles. They share 3 edges and the vertices are identified (more on: Triangulation of Torus).
If you would like to calculate the homology goup by using a CW complex, you just take a 0-cell and attach two 1-cells. Then you attach one 2-cell. The following video gives you an impression how to glue the 2-cell, although this video is not originally made for illustrating this: https://www.youtube.com/watch?v=nLcr-DWVEto
To get an intuitive understanding of the subject, I recommend you to watch the videos of Wildberger: https://www.youtube.com/watch?v=Uq4dTjHfLpI