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Cellular Homology
In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex with n-skeleton, ''n''-skeleton X_ , the cellular-homology modules are defined as the homology groups ''Hi'' of the cellular chain complex : \cdots \to (X_,X_) \to (X_,X_) \to (X_,X_) \to \cdots, where X_ is taken to be the empty set. The group : (X_,X_) is Free module, free abelian, with generators that can be identified with the n -cells of X . Let e_^ be an n -cell of X , and let \chi_^: \partial e_^ \cong \mathbb^ \to X_ be the attaching map. Then consider the composition : \chi_^: \mathbb^ \, \stackrel \, \partial e_^ \, \stackrel \, X_ \, \stackrel \, X_ / \left( X_ \setminus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus G Surface
In mathematics, a genus ''g'' surface (also known as a ''g''-torus or ''g''-holed torus) is a surface formed by the connected sum of ''g'' many tori: the interior of a disk is removed from each of ''g'' many tori and the boundaries of the ''g'' many disks are identified (glued together), forming a ''g''-torus. The genus of such a surface is ''g''. A genus ''g'' surface is a two-dimensional manifold. The classification theorem for surfaces states that every compact connected two-dimensional manifold is homeomorphic to either the sphere, the connected sum of tori, or the connected sum of real projective planes. Definition of genus 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 ''χ'', via the relationship ''χ''&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albrecht Dold
Albrecht Dold (5 August 1928 – 26 September 2011) was a German mathematician specializing in algebraic topology who proved the Dold–Thom theorem, the Dold–Kan correspondence, and introduced Dold manifolds, Dold–Puppe stabilization, and Dold fibrations. Life Albrecht Dold was born in Triberg, and studied mathematics and physics at Heidelberg University, earning a Ph.D. degree in 1954 under the direction of Herbert Seifert. He visited the Institute for Advanced Study in Princeton in 1956–58, and taught at Columbia University in 1960–62 and at the University of Zürich in 1962–63. In 1963 he returned to Heidelberg, where he stayed most of his career, till his retirement in 1996. Dold's work in algebraic topology, in particular, his work on fixed-point theory has made him known in economics as well as mathematics. His book "Lectures on Algebraic Topology" is a standard reference among economists as well as mathematicians. He had 19 doctoral students, including Mó ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Homology
In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for topological pair, pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace. Definition Given a subspace A\subseteq X, one may form the short exact sequence :0\to C_\bullet(A) \to C_\bullet(X)\to C_\bullet(X) /C_\bullet(A) \to 0 , where C_\bullet(X) denotes the singular chains on the space ''X''. The boundary map on C_\bullet(X) descends to C_\bullet(A) and therefore induces a boundary map \partial'_\bullet on the quotient. If we denote this quotient by C_n(X,A):=C_n(X)/C_n(A), we then have a complex :\cdots\longrightarrow C_n(X,A) \xrightarrow C_(X,A) \longrightarrow \cdots . By definition, the th relative homology group of the pair of spaces (X,A) is :H_n(X,A) := \ker\partial'_n/\operatorname\par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betti Number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if H_n(X) \cong 0 then b_n(X) = 0, if H_n(X) \cong \mathbb then b_n(X) = 1, if H_n(X) \cong \mathbb \oplus \mathbb then b_n(X) = 2, if H_n(X) \cong \mathbb \oplus \mathbb\oplus \mathbb then b_n(X) = 3, etc. Note that only the ranks of infinite groups are considered, so for example if H_n(X) \cong \mathbb^k \oplus \mathbb/(2) , where \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extraordinary Homology Theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah–Hirzebruch Spectral Sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, it relates the generalized cohomology groups : E^i(X) with 'ordinary' cohomology groups H^j with coefficients in the generalized cohomology of a point. More precisely, the E_2 term of the spectral sequence is H^p(X;E^q(pt)), and the spectral sequence converges conditionally to E^(X). Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where E=H_. It can be derived from an exact couple that gives the E_1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E. In detail, assume X to be the total space of a Serre fibration with fibre F and base space B. The filtration of B by its n-skeletons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Projective Space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the ''complex'' lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (''n''+1)-dimensional complex vector space. The space is denoted variously as P(C''n''+1), P''n''(C) or CP''n''. When , the complex projective space CP1 is the Riemann sphere, and when , CP2 is the complex projective plane (see there for a more elementary discussion). Complex projective space was first introduced by as an instance of what was then known as the "geometry of position", a notion originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. Sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction As with all projective spaces, RP''n'' is formed by taking the quotient of under the equivalence relation for all real numbers . For all ''x'' in one can always find a ''λ'' such that ''λx'' has norm 1. There are precisely two such ''λ'' differing by sign. Thus RP''n'' can also be formed by identifying antipodal points of the unit ''n''-sphere, ''S''''n'', in R''n''+1. One can further restrict to the upper hemisphere of ''S''''n'' and merely identify antipodal points on the bounding equator. This shows that RP''n'' is also equivalent to the closed ''n''-dimensional disk, ''D''''n'', with antipodal points on the boundary, , identified. Low-dimensional examples * RP1 is called the real projective line, which is topologically equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Polygon
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following: *the Riemann sphere, *the complex plane, *the unit disk ''D'' or equivalently the upper half-plane ''H''. In the first case of genus zero, the surface is conformally equivalent to the Riemann sphere. In the second case of genus one, the surface is conformally equivalent to a torus C/Λ for some lattice Λ in C. The fundamental polygon of Λ, if assumed convex, may be taken to be either a period parallelogram or a centrally symmetric hexagon, a result first proved by Fedorov in 1891. In the last case of genus ''g'' > 1, the Riemann surface is conformally equivalent to ''H''/Γ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
