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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.

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"Combinatorial" refers to the use of triangulations ( en.wikipedia.org/wiki/Triangulation_(geometry) , en.wikipedia.org/wiki/Simplicial_homology ). It doesn't refer to a counting problem; rather it refers to the use of combinatorial objects to model topological ones. I am sure others could give a more complete answer though. –  Qiaochu Yuan Oct 7 '11 at 16:09
    
But I can say Combinatorial topology is a type algebraic topology that uses combinatorial methods,as take Simplicial complex ,thats where your homology arises ,we can study the homology arising from simplicial complexes ,and Simplicial homology is a type of combinatorial construction of Algrebraic topology,thats what i understood,but by the way the i noticed something funny ,Combinatorial Topology and Topological Combinatorics are different,i mean they are Anagrams(words which are arranged in other sense produce different meaning) –  Iyengar Oct 7 '11 at 17:36
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3 Answers

up vote 14 down vote accepted

The transformation of intuitive visual topology, such as the homology and fundamental group of two and three-dimensional spaces, into a rigorous subject, was first done using triangulations. A manifold would be triangulated, definition and computations made relative to a triangulation, and the answer shown to be independent of the triangulation. I think Lefschetz called this part of topology "combinatorial analysis situs".

In this approach a critical question is whether every continuous or smooth manifold has a triangulation and whether all triangulations of the same space are equivalent. Due to the history of basing everything on triangulations this matter was designated the Hauptvermutung (Main Conjecture).

At the time the problem was formulated there may not have been complete clarity about how to define a manifold, or a recognition of the possible difference between smooth and continuous manifolds. In modern terms the triangulation approach is part of Piecewise Linear topology and the Hauptvermutung asks about existence and uniqueness of PL structures on a topological manifold, and their relation to smooth structures. The original hope, for existence and uniqueness of triangulation, proved to be true in dimensions up to 3, and false in higher dimensions.

There is a part of combinatorics that studies abstract simplicial complexes as systems of finite subsets of a finite or discrete set. This is inspired by topology but appeared after topology had shed its conceptual dependence on triangulations.

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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.

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Thanks! thats a very good answer! –  Dinesh Nov 8 '11 at 17:58
    
+1 for both a very good complimentary answer to mine and mentioning one of my favorite old-school topology textbooks that straddles the line between classical combinatorial topology and modern abstract topology. A worthy addition to any mathematical library. –  Mathemagician1234 Nov 24 '11 at 2:03
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Combinatorial topology is the older name for algebraic topology when all topological problems were expressed, set up and solved in Euclidean space of dimensions 1,2 and 3. In such spaces, all topological invariants-such as the fundamental group-can be expressed combinatorially via simplexes and related objects. This approach has the large advantage of being highly visual and geometric. After the second World War, topology began to deal with much more abstract constructions in higher spaces and after that, the combinatorial viewpoint became much less useful and algebraic structures such as homology groups and functors took precedent.

An outstanding presentation of combinatorial topology via the historical development of topology before World War II can be found in John Stillwell's Classical Topology And Combinatorial Group Theory.

Addendum: Here's a good way to think of it: Combinatorial topology is to modern algebraic topology what classical real analysis/calculus is to point set topology. Classical real analysis is the special case where the underlying topological space is Euclidean space with the usual metric topology. Point set topology generalizes these concepts to arbitrary topological spaces where the topology does not even need to be generated by a metric. As a result, the traditional methods of determining open balls and continuous maps in calculus are nearly useless in general non-metric topological spaces and abstract set-theoretic methods need to be used.

In the same manner, combinatorial topology deals with the "translation" of geometrical constructions (simplexes,CW complexes,etc.) in low dimensions to algebraic constructs (i.e. the fundamental group, homotopy groups,etc.) In dimensions less then or equal to n =3, the geometric constructs are not only easily visualizable, but the combinatorial data and the algebraic data contain exactly the same topological information-since any geometric construct is decomposable by a triangulation that uniquely determines its fundamental group-and as a result,they are effectively interchangeable. In higher dimensions on arbitrary topological spaces, this interchangeability is false and many problems are not visualizable at all (for example, the "hard" Whitney theorum and spectral sequences). As a result, modern topology uses category theory to transform intractably abstract geometric problems in the category TOP into more precise algebraic problems by functors into algebraic categories such as GROUP, which in many cases are solvable.

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thanks for the reference..funny, the impact of world war on topology :D –  Dinesh Oct 7 '11 at 20:18
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