In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. It has come over time to be almost synonymous with low-dimensional topology, concerning in particular manifolds of four or fewer dimensions. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... Trefoil knot, the simplest non-trivial knot. ... In mathematics, the braid group on n strands, denoted by Bn, is a certain group which has a nice geometrical representation and in a sense generalizes the symmetric group Sn. ...

In the rapid development of topology after 1945, a distinction was drawn between the fields of algebraic topology typified by homotopy theory, geometric topology with the Poincaré conjecture as its biggest unsolved problem, and differential topology as the study mostly of differential structures, with Morse theory and transversality as their natural techniques. These fields all rested on general topology, which was the study of the general topological space. This classification would come to seem more artificial, with the passing of years. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold. ... A Morse function is also an expression for an anharmonic oscillator In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ... Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the opposite of tangency, and plays a role in general position. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

A number of advances starting in the 1960s had the effect of changing geometric topology. The solution by Smale, in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics. Vaughan Jones' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics. In 2002 Grigori Perelman announced a proof of the three-dimensional Poincare conjecture, using Richard Hamilton's Ricci flow: an idea belonging to the field of geometric analysis. Stephen Smale (born July 15, 1930) is an American mathematician and winner of the Fields Medal in 1966. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... In mathematics, specifically in topology, surgery theory is the name given to a collection of techniques used to produce one manifold from another in a controlled way. ... William Thurston William Paul Thurston (born October 30, 1946) is an American mathematician. ... The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a two-sided incompressible surface. ... Vaughan Frederick Randal Jones (born 31 December 1952) is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. ... This article needs cleanup. ... For album titles with the same name, see 2002 (album). ... Grigori Yakovlevich Perelman (Russian: ), born 13 June 1966 in Leningrad, USSR (now St. ... Richard Hamilton is the name of: Richard Hamilton (artist), a British painter and collage artist Richard Hamilton (basketball), a player with the Detroit Pistons of the National Basketball Association Richard Hamilton (professor), Professor of Mathematics at Columbia University Richard Hamilton (actor) [1] This is a disambiguation page: a list of... In differential geometry, Ricci flow is the flow of Riemannian metrics given by the equation where g is the metric and Ric is the Ricci curvature. ... This article or section is in need of attention from an expert on the subject. ...

Overall, this progress has led to better integration of the field into the rest of mathematics.

See also

• list of geometric topology topics

This is a list of geometric topology topics, by Wikipedia page, organized roughly by dimension. ...

External links

• Rob Kirby's Problems in Low-Dimensional Topology -gzipped postscript file (1.4MB)
• Mark Brittenham's links to low dimensional topology - lists of homepages, conferences, etc.