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Lens Space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their boundaries. Often the 3-sphere and S^2 \times S^1, both of which can be obtained as above, are not counted as they are considered trivial special cases. The three-dimensional lens spaces L(p,q) were introduced by Heinrich Tietze in 1908. They were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type. J. W. Alexander in 1919 showed that the lens spaces L(5;1) and L(5;2) were not homeomorphic even though they have isomorphic fundamental groups and the same homology, though they do not have th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lens Space L(2;5) Side View
A lens is a transmissive optics, optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a #Compound lenses, compound lens consists of several simple lenses (''elements''), usually arranged along a common Optical axis, axis. Lenses are made from materials such as glass or plastic, and are Grinding (abrasive cutting), ground and Polishing, polished or Molding (process), molded to a desired shape. A lens can focus light to form an image, unlike a Prism (optics), prism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses. Lenses are used in various imaging devices like telescopes, binoculars and cameras. They are also used as visual aids in glasses to correct defects of vision such as Near-sightedness, myopia and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Symmetric Space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (''M'', ''g'') is said to be symmetric if and only if, for each point ''p'' of ''M'', there exists an isometry of ''M'' fixing ''p'' and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glen Bredon
Glen Eugene Bredon (August 24, 1932 in Fresno, California – May 8, 2000, in North Fork, California) was an American mathematician who worked in the area of topology. Education and career Bredon received a bachelor's degree from Stanford University in 1954 and a master's degree from Harvard University in 1955. In 1958 he wrote his PhD thesis at Harvard (''Some theorems on transformation groups'') under the supervision of Andrew M. Gleason. Starting in 1960 he worked as a professor at the University of California, Berkeley and since 1969 at Rutgers University, until he retired in 1993, after which he moved to North Fork, California. From 1958 to 1960 and 1966/67 he was at the Institute for Advanced Study. The Bredon cohomology of topological spaces under action of a topological group is named after him. In the late 1980s, he wrote the program DOS.MASTER for Apple II computers. He is the author of the programs Merlin (a macro assembler) and ProSel for Apple machines. Pers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical 3-manifold
In mathematics, a spherical 3-manifold ''M'' is a 3-manifold of the form :M=S^3/\Gamma where \Gamma is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds or Clifford-Klein manifolds. Properties A spherical 3-manifold S^3/\Gamma has a finite fundamental group isomorphic to Γ itself. The elliptization conjecture, proved by Grigori Perelman, states that conversely all compact 3-manifolds with finite fundamental group are spherical manifolds. The fundamental group is either cyclic, or is a central extension of a dihedral, tetrahedral, octahedral, or icosahedral group by a cyclic group of even order. This divides the set of such manifolds into 5 classes, described in the following sections. The spherical manifolds are exactly the manifolds with spherical geometry, one of the 8 geometries of Thurston's geometrization conjecture. Cyclic ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massey Product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist. Massey triple product Let a,b,c be elements of the cohomology algebra H^*(\Gamma) of a differential graded algebra \Gamma. If ab=bc=0, the Massey product \langle a,b,c\rangle is a subset of H^n(\Gamma), where n=\deg(a)+\deg(b)+\deg(c)-1. The Massey product is defined algebraically, by lifting the elements a,b,c to equivalence classes of elements u,v,w of \Gamma, taking the Massey products of these, and then pushing down to cohomology. This may result in a well-defined cohomology class, or may result in indeterminacy. Define \bar u to be (-1)^u. The cohomology class of an element u of \Gamma will be denoted by /math>. The Massey triple product of three cohomology classes is defined by : \langle rangle = \. The Massey product of three cohomology classes is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Configuration Space (mathematics)
In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positions in a topological space. More specifically, configuration spaces in mathematics are particular examples of configuration spaces in physics in the particular case of several non-colliding particles. Definition For a topological space X, the ''n''th (ordered) configuration space of X is the set of ''n''-tuples of pairwise distinct points in X: :\operatorname_n(X):=\prod^n X \smallsetminus \. This space is generally endowed with the subspace topology from the inclusion of \operatorname_n(X) into X^n. It is also sometimes denoted F(X, n), F^n(X), or \mathcal^n(X). There is a natural action of the symmetric group S_n on the points in \operatorname_n(X) given by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial. Definition Let ''K'' be a knot in the 3-sphere. Let ''X'' be the infinite cyclic cover of the knot complement of ''K''. This covering can be obtained by cutting the knot complement along a Seifert surface of ''K'' and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a covering transformation ''t'' acting on ''X''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Obstruction
In mathematics, specifically in surgery theory, the surgery obstructions define a map \theta \colon \mathcal (X) \to L_n (\pi_1 (X)) from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when n \geq 5: A degree-one normal map (f,b) \colon M \to X is normally cobordant to a homotopy equivalence if and only if the image \theta (f,b)=0 in L_n (\mathbb pi_1 (X). Sketch of the definition The surgery obstruction of a degree-one normal map has a relatively complicated definition. Consider a degree-one normal map (f,b) \colon M \to X. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve (f,b) so that the map f becomes m-connected (that means the homotopy groups \pi_* (f)=0 for * \leq m) for high m. It is a consequence of Poincaré duality that if we can achieve this for m > \lfloor n/2 \rfloor then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Classes
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Definition Let ''G'' be a topological group, and for a topological space X, write b_G(X) for the set of isomorphism classes of principal ''G''-bundles over X. This b_G is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map f\colon X\to Y to the pullback operation f^*\colon b_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Homotopy
In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions (inverses of collapses), and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map. The obstruction to a homotopy equivalence being a simple homotopy equivalence is the Whitehead torsion, \tau(f). A homotopy theory that studies simple-homotopy types is called simple homotopy theory. See also * Discrete Morse theory Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology com ... References * Homotopy theory Equivalence (mathematics) {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PL Homeomorphism
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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