I remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting problem. I have seen how the fundamental group is constructed and the techniques of covering spaces to find the fundamental group of some spaces. In this process I could never see any counting problem involved in it. All that I can see is that the Fundamental Groups( so are the homotopy and homology groups) are topological invariants of certain space and describe its structure(modulo some restrictions).
Question : Can any one please explain(preferably through illustrative examples) how this subject has some "Combinatorial" flavor in it?
Background: Fundamental Groups, Covering spaces and some hand waving knowledge of homology groups.
Note : All that I can imagine is that by Cayleys theorem fundamental group can be seen as a subgroup of some permutation group and since permutation group is very much a combinatorial object, it is known to be so. But it is just an imagination and I believe there is much more to it.
Best Answer
When Poincaré first envisioned algebraic topology, he envisioned it as a study of smooth manifolds under the equivalence relation if diffeomorphism [Analysis situs, pages 196-198]. Lefshetz [Topology, Amer. Math. Soc. Colloq. Publ 12 (1930), page 361] wrote that Poincaré had tried to develop the subject along `analytic' lines, but had turned instead to combinatorial methods because the analytic approach failed for example in the Poincaré duality theorem.
Algebraic topology developed in the PL category (Combinatorial Topology), because it was believed that this would give a useful avenue of attack on the differentiable case. Great algebraic topologists of the early 20th century (Reidemeister, Seifert, Schubert$\thinspace\ldots$) all worked with triangulated PL manifolds, and wrote good, precise, rigourous papers which are still valuable today (I think that Schubert's Topologie is one of the greatest topology textbooks ever written). They were gluing together, and subdividing, finite collections of simplices; their group theory was combinatorial; and the subject as a whole had a highly combinatorial flavour to it. And it was great! Everything was explicit, and there was no need for fudgy handwavy `corners can be rounded' type arguments to be thrown around. In my opinion, simplicial complexes continue to be the best setting to work explicitly with linking forms, for example.
In the 1950's and 1960's, with work of Smale, Thom, Milnor, Hirsch, and others, honest smooth algebraic topology became possible, and the relationship between PL and smooth categories was clarified. And after that, people began switching back and forth at will when it was possible to do so, and, with the basic groundwork for algebraic topology established in both categories, the combinatorial flavour of the subject became dulled. Combinatorial Group Theory went off and became its own subject, and the majority of topologists no longer saw the need to mess about with explicit triangulations of manifolds- they just worked directly with invariants of the chain complex. And CW complexes became used instead of simplicial complexes, for example because the dual cell subdivision of a simplicial complex need no longer be a simplicial complex.
But "combinatorial topology" in its former sense still very much exists. An it's not going to go away. To programme topology into a computer for example, you need an explicit triangulation, and the work is all combinatorial and PL. See for example Matveev's Algorithmic topology and classification of 3-manifolds. The constructivist argument would be that `real world' manifolds (whatever that means) are PL.