Algebraic topology

Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.

The method of algebraic invariants

The goal is to take topological spaces, and further categorize or classify them. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was contructed from simpler ones. The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants: for example by mapping them to groups, which have a great deal of manageable structure, in a way that respects the relation of homeomorphism of spaces.

Two major ways in which this can be done are through fundamental groups, or more general homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space; but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation.

Homology and cohomology groups, on the other hand, are abelian, and in many important cases finitely generated. Finitely generated abelian groups can be completely classified and are particularly easy to work with.

Results on homology

Several useful results follow immediately from working with finitely generated abelian groups. 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. If an n-th homology group of a simplicial complex has torsion, then the complex is nonorientable (query this). Thus, a great deal of topological information is encoded in the homology of a given topological space.

Beyond simplicial homology, one can use the differential structure of smoothmanifolds via de Rham cohomology , or Cech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question. De Rham 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 De Rham cohomology. {That would be a compact oriented manifold then, to use Poincare duality.)

Setting in category theory

In general, all constructions of algebraic topology are functorial: the notions of category, functor and natural transformation originated here. Fundamental groups, homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups; a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.

The problems of algebraic topology

The most celebrated geometric open problem in algebraic topology is the Poincaré conjecture. The field of homotopy theory contains many mysteries, in particular the right way to describe the homotopy groups of spheres.

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