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Lickorish–Wallace Theorem
In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted. The theorem was proved in the early 1960s by W. B. R. Lickorish and Andrew H. Wallace, independently and by different methods. Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism of a closed orientable surface is generated by Dehn twists along 3''g'' − 1 specific simple closed curves in the surface, where ''g'' denotes the genus of the surface. Wallace's proof was more general and involved adding handles to the boundary of a higher-dimensional ball. A corollary of the theorem is that every closed, orientable 3-manifold bounds a simply-connected compact 4-manifold In mathematics, a 4-manifold is a 4-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In an algebraic structure such as a group, a ring, or vector space, an ''automorphism'' is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f: X\to X is an automorphism if there is a morphism g: X\to X such that g\circ f= f\circ g = \operatorname _X, where \operatorname _X is the identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifolds
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Principles Definition A topological space M is a 3-manifold if it is a second-countable Hausdorff space and if every point in M has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of implies that the homeomorphism type of the manifold only depends on this intersection form, and on a \Z/2\Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus (mathematics)
In mathematics, genus (: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic \chi, via the relationship \chi=2-2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads \chi=2-2g-b. In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dehn Twist
In geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ..., a branch of mathematics, a Dehn twist is a certain type of homeomorphism, self-homeomorphism of a Surface (topology), surface (two-dimensional manifold). Definition Suppose that ''c'' is a curve, simple closed curve in a closed, Orientability, orientable surface ''S''. Let ''A'' be a tubular neighborhood of ''c''. Then ''A'' is an Annulus (mathematics), annulus, homeomorphic to the Cartesian product of a circle and a unit interval ''I'': :c \subset A \cong S^1 \times I. Give ''A'' coordinates (''s'', ''t'') where ''s'' is a complex number of the form e^ with \theta \in [0, 2\pi], and . Let ''f'' be the map from ''S'' to itself which is the identity outside of ''A'' and inside ''A'' we have :f(s, t) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world. In general In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lickorish Twist Theorem
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold). Definition Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ''A'' be a tubular neighborhood of ''c''. Then ''A'' is an annulus, homeomorphic to the Cartesian product of a circle and a unit interval ''I'': :c \subset A \cong S^1 \times I. Give ''A'' coordinates (''s'', ''t'') where ''s'' is a complex number of the form e^ with \theta \in , 2\pi and . Let ''f'' be the map from ''S'' to itself which is the identity outside of ''A'' and inside ''A'' we have :f(s, t) = \left(se^, t\right). Then ''f'' is a Dehn twist about the curve ''c''. Dehn twists can also be defined on a non-orientable surface ''S'', provided one starts with a 2-sided simple closed curve ''c'' on ''S''. Example Consider the torus represented by a fundamental polygon with edges ''a'' and ''b'' :\mathbb^2 \cong \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Principles Definition A topological space M is a 3-manifold if it is a second-countable Hausdorff space and if every point in M has a neighbourhood (mathematics), neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, Piecewise linear manifold, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew H
Andrew is the English form of the given name, common in many countries. The word is derived from the , ''Andreas'', itself related to ''aner/andros'', "man" (as opposed to "woman"), thus meaning "manly" and, as consequence, "brave", "strong", "courageous", and "warrior". In the King James Bible, the Greek "Ἀνδρέας" is translated as Andrew. Popularity In the 1990s, it was among the top ten most popular names given to boys in English-speaking countries. Australia In 2000, the name Andrew was the second most popular name in Australia after James. In 1999, it was the 19th most common name, while in 1940, it was the 31st most common name. Andrew was the first most popular name given to boys in the Northern Territory in 2003 to 2015 and continuing. In Victoria, Andrew was the first most popular name for a boy in the 1970s. Canada Andrew was the 20th most popular name chosen for male infants in 2005. Andrew was the 16th most popular name for infants in British Columbia i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior of a 3-sphere is a 4-ball. It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, Four-dimensional space, 4-dimensional space () such that :\sum_^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted : :S^3 = \left\. It is often convenient to r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
