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 constructed from simpler ones. 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 of spaces.

Related Topics:
Combinatorial topology - Groups - Homeomorphism

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

Related Topics:
Fundamental group - Homotopy theory - Homology - Cohomology - Nonabelian - Simplicial complex - 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.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~