home | alphabetical index | |||||

## Algebraic topologyAlgebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.## The method of algebraic invariantsThe 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
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. { ## 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 ## The problems of algebraic topologyThe 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. ## External links | |||||

copyright © 2004 FactsAbout.com |