Algebraic topology

related topics
{math, number, function}
{group, member, jewish}
{style, bgcolor, rowspan}

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, though usually most classify up to homotopy equivalence.

Although algebraic topology primarily uses algebra to study topological problems, 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.

Full article ▸

related documents
Generalized Riemann hypothesis
Isomorphism theorem
Monotonic function
Conjunctive normal form
Axiom of regularity
Tree (data structure)
Differential topology
Cartesian product
Knight's tour
Mean value theorem
Laurent series
Kolmogorov space
Spectrum of a ring
Product topology
Diophantine set
Theory of computation
De Morgan's laws
Lebesgue measure
Goodstein's theorem
Carmichael number
Group representation
Local ring
Least common multiple
Cauchy-Riemann equations
Convex set
Automata theory