The goal is to take topological spaces, and further categorize them by mapping them to groups, which have a great deal of structure. Two major ways in which this can be done are through fundamental groups, or homotopy, and through homolog/cohomology groups. Fundamental groups give us crucial information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. Homology/Cohomology groups, on the other hand, are abelian, and in many important cases even finitely generated. Finitely generated abelian groups can be completely classified and are particularly easy to work with.
Several nice results follow immediately from working with finitely generated abelian groups. If an n-th homology group of a simplicial complex has torsion, then the complex is nonorientable. The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic. Thus, a great deal of topological information is encoded in the homology of a given topological space.
Beyond simplicial homology, one can use De Rham cohomology to investigate the differential structure of manifolds, or Cech or Sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question. DeRham showed that all of these approaches were interrelated and that the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through DeRham cohomology.
In general, all constructions of algebraic topology are functorial (and the notions of category, functor and natural transformation originated here). In particular, fundamental groups, homology and cohomology groups are invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups.
The most important open problem in algebraic topology is the Poincaré conjecture.
- Allen Hatcher's book Algebraic Topology is available free in PDF and PostScript formats: http:/www.math.cornell.edu/~hatch/ATpage.html