Search results
Results from the WOW.Com Content Network
Algebraic topology is a branch of mathematics that 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 ...
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
Chain (algebraic topology) Betti number; Euler characteristic. Genus; Riemann–Hurwitz formula; Singular homology; Cellular homology; Relative homology; Mayer–Vietoris sequence; Excision theorem; Universal coefficient theorem; Cohomology. List of cohomology theories; Cocycle class; Cup product; Cohomology ring; De Rham cohomology; Čech ...
5 Examples. 6 Uniform spaces. 7 Metric spaces. ... 11 Combinatorial topology. 12 Foundations of algebraic topology. 13 Topology and algebra. 14 See also. Toggle the ...
The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. This is a list of topology topics. See also: Topology glossary; List of topologies; List of general topology topics; List of geometric topology topics
In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by
The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group ...
Category theory is the language of modern algebra, and has been widely used in the study of algebraic geometry and topology. It has been noted that "the key observation of [ 10 ] is that the persistence diagram produced by [ 8 ] depends only on the algebraic structure carried by this diagram."