What is the point of triangulating topological spaces?

Question

In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra?

Wikipedia says homology and cohomology groups can be computed with simplicial homology and cohomology theories, as opposed to more complicated homology and cohomology theories.

I don't really understand this as my grasp of topology and geometry is quite weak. I would appreciate it if someone could answer this in a simple manner (examples of where triangulations are useful in real life applications would be awesome!!)

Answer 1

If you only know about singular homology, which is defined directly for topological spaces, most explicit calculations are very hard. Having a triangulation as a $\Delta$-complex, simplicial-complex or even just a CW-complex reduces the topological space to a finite set of gluing data of simplices (or spheres in the CW-case). This data can be put into a computer and the computer can tell you all about the homology and cohomology groups of the complex. Then you have theorems telling you that singular homology and say simplicial homology are isomorphic, so you really learned something about your triangulated space! So the main advantage of triangulations in algebraic topology is being able to explicitly compute a lot of algebraic invariants of a topological space.

Answer 2

Suppose you are interested in algorithmic aspects of topological manifolds, i.e. being able to determine if two manifolds are homeomorphic or not. In order to solve this problem, you first have to represent the manifold in the form of some computable data. Triangulation makes it possible. On the other hand, if all what you know is that your manifold is given as the zero-level set of some smooth function, it is (in general) very much unclear how to describe your manifold in a computable form.

In the case of 3-dimensional manifolds, at least theoretically, once you have a triangulation, you can then determine the connected sum decomposition of the manifold (Haken-Kneser algorithm), then split it along incompressible tori, etc.