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I am currently pursuing a course in basic homology theory and i am finding it really difficult to find the triangulation of spaces. I know that a triangulation of a topological space $X$ is a simplicial complex $K$, homeomorphic to $X$, together with a homeomorphism from $K$ onto $X$.

For some simple spaces, say, for $X=[0,1]$ i can identify it as a simplicical complex $K=\{<0>,<1>,<0,1>\}$ and then clearly $|K|$ is isomorphic to $X$ by the identity map.

Similarly, for $\Bbb D^2$ = {x $\in \Bbb R^2$: ||x||$\le$1}, now, in the book it says that its triangulation is a $2$-simplex. But what will be a homeomorphism in this case.

Similarly, given a torus, double torus, a cylinder, tetrahedron etc. How exactly do we find out its triangulation? Is it necessary to find out the homeomorphism in every single case or are there any other easier methods to find out the triangulations?

Please explain in detail the method used to identify triangulations of given spaces. Also, if possible mention some reference texts which have enough examples on this topic.Also, does any Algebraic Topology textbook covers in depth on Triangulation and Euler Characteristic? Most of the books I have read thus far (Munkres, Hatcher, etc.) do mention it but only briefly.

thanks a lot!!...

Abhishek Shrivastava
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Are you convinced that these are homeomorphisms? Really proving that something is a homeomorphism to its image can be a pain. One usually resolves to making some drawings... The point is that the images of simplices in a triangulation can only intersect in lower dimensional simplices. Examples a plenty on this site.

However, we have a nice lemma if you want to really prove that the maps are homeomorphisms:

Let $X$ compact, and $Y$ a Hausdorff topological space. Suppose $f:X\rightarrow Y$ is a continuous map that is injective. Then $f$ is a homeomorphism to its image.

See https://proofwiki.org/wiki/Continuous_Bijection_from_Compact_to_Hausdorff_is_Homeomorphism.

Thomas Rot
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