Algebraic topology

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

The problems of algebraic topology

Classic applications of algebraic topology include:

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• 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.

The most celebrated geometric open problem in algebraic topology is the Poincaré conjecture, which may have been resolved by Grigori Perelman. The field of homotopy theory contains many mysteries, most famously the right way to describe the homotopy groups of spheres.

Related Topics:
Poincaré conjecture - Grigori Perelman - Homotopy theory - Homotopy groups of spheres

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