Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy equivalence.

Although algebraic topology primarily uses algebra to study topological problems, the converse, using topology to solve algebraic problems, is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

The method of algebraic invariants
An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW-complex). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove.

Two major ways in which this can be done are through fundamental groups, or more generally 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 are completely classified and are particularly easy to work with.

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 and 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, but their associated morphisms also correspond — 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.

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. As another example, the top-dimensional integral homology group of a closed manifold detects orientability: this group is isomorphic to either the integers or 0, according as the manifold is orientable or not. Thus, a great deal of topological information is encoded in the homology of a given topological space.

Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth manifolds via de Rham cohomology, or Čech 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, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. This was extended in the 1950s, when Eilenberg and Steenrod generalized this approach. They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms (e.g., a weak equivalence of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.

A new approach uses a functor from filtered spaces to crossed complexes  defined directly and homotopically using relative homotopy groups; a higher homotopy van Kampen theorem proved for this functor enables basic results in algebraic topology, especially on the border between homology and homotopy, to be obtained without using singular homology or simplicial approximation. This approach is also called non abelian algebraic topology, and generalises to higher dimensions ideas coming from the fundamental group.

Applications of algebraic topology
Classic applications of algebraic topology include:
 * The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point.
 * The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n=2, this is sometimes called the "hairy ball theorem".)
 * The Borsuk-Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points.
 * Any subgroup of a free group is free. This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X. The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X; but every such Y is again a graph. Therefore its fundamental group H is free.
 * Topological combinatorics

Notable algebraic topologists

 * Frank Adams
 * Karol Borsuk
 * Luitzen Egbertus Jan Brouwer
 * William Browder
 * Nicolas Bourbaki
 * Henri Cartan
 * Otto Hermann Künneth
 * Samuel Eilenberg
 * Peter Freyd
 * Alexander Grothendieck
 * Friedrich Hirzebruch
 * Heinz Hopf
 * Michael J. Hopkins
 * Witold Hurewicz
 * Egbert van Kampen
 * Saunders Mac Lane
 * J.P. May
 * John Coleman Moore
 * Sergei Petrovich Novikov
 * Lev Pontryagin
 * Daniel Quillen
 * Jean-Pierre Serre
 * Norman Steenrod
 * Dennis Sullivan
 * René Thom
 * Hassler Whitney
 * J. H. C. Whitehead

Important theorems in algebraic topology

 * Borsuk-Ulam theorem
 * Brouwer fixed point theorem
 * Cellular approximation theorem
 * Eilenberg–Zilber theorem
 * Hurewicz theorem
 * Kunneth theorem
 * Poincaré duality theorem
 * Universal coefficient theorem
 * Van Kampen's theorem
 * Generalized van Kampen's theorems
 * Higher homotopy, generalized van Kampen's theorem
 * Whitehead's theorem