In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology.
In mathematics, topology concerns with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
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.
In mathematics, the braid group on n strands, also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids, and whose group operation is composition of braids. Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids ; in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation ; and in monodromy invariants of algebraic geometry.
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.
In knot theory, a branch of mathematics, a knot or link in the 3-dimensional sphere is called fibered or fibred if there is a 1-parameter family of Seifert surfaces for , where the parameter runs through the points of the unit circle , such that if is not equal to then the intersection of and is exactly .
In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant, is a knot invariant that can be extended to an invariant of certain singular knots that vanishes on singular knots with m + 1 singularities and does not vanish on some singular knot with 'm' singularities. It is then said to be of type or order m.
Joan Sylvia Lyttle Birman is an American mathematician, specializing in low-dimensional topology. She has made contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical systems. Birman is research professor emerita at Barnard College, Columbia University, where she has been since 1973.
In the mathematical field of low-dimensional topology, a clasper is a surface in a 3-manifold on which surgery can be performed.
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.
In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent knots or links. The conditions are stated in terms of the group structures on braids.
Vladimir Georgievich Turaev is a Russian mathematician, specializing in topology.
The Joan & Joseph Birman Research Prize in Topology and Geometry was established in 2013 and is a prize given every other year by the Association for Women in Mathematics to an outstanding young female researcher in topology or geometry. The prize fund for the award was established by a donation from Joan Birman and her husband, Joseph Birman.
Introduction to 3-Manifolds is a mathematics book on low-dimensional topology. It was written by Jennifer Schultens and published by the American Mathematical Society in 2014 as volume 151 of their book series Graduate Studies in Mathematics.
