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Homological Connectivity
In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. Definitions Background ''X'' is ''homologically-connected'' if its 0-th homology group equals Z, i.e. H_0(X)\cong \mathbb, or equivalently, its 0-th reduced homology group is trivial: \tilde(X)\cong 0. * For example, when ''X'' is a graph and its set of connected components is ''C'', H_0(X)\cong \mathbb^ and \tilde(X)\cong \mathbb^ (see graph homology). Therefore, homological connectivity is equivalent to the graph having a single connected component, which is equivalent to graph connectivity. It is similar to the notion of a connected space. ''X'' is ''homologically 1-connected'' if it is homologically-connected, and additionally, its 1-th homology group is trivial, i.e. H_1(X)\cong 0. * For example, when ''X'' is a connected graph with vertex-set ''V'' and edge-set ''E'', H_1(X) \cong \mathbb^. Therefore, homological 1-connectivity is equivalent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
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. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Homology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Homology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are ''n''- simplices, the ''n''-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a ''triangulation'' of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead). Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simpl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meshulam's Game
In graph theory, Meshulam's game is a game used to explain a theorem of Roy Meshulam related to the homological connectivity of the independence complex of a graph, which is the smallest index ''k'' such that all reduced homological groups up to and including ''k'' are trivial. The formulation of this theorem as a game is due to Aharoni, Berger and Ziv. Description The game-board is a graph ''G.'' It is a zero-sum game for two players, CON and NON. CON wants to show that I(''G''), the independence complex of ''G'', has a high connectivity; NON wants to prove the opposite. At his turn, CON chooses an edge ''e'' from the remaining graph. NON then chooses one of two options: * ''Disconnection'' – remove the edge ''e'' from the graph. * ''Explosion'' – remove both endpoints of ''e'', together with all their neighbors and the edges incident to them. The score of CON is defined as follows: * If at some point the remaining graph has an isolated vertex, the score is infinity; * O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopical Connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of ''n''-connectedness generalizes the concepts of path-connectedness and simple connectedness. An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is ''n''-connected (or ''n''-simple connected) if its first ''n'' homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is ''n''-connected if it is an isomorphism "up to dimension ''n,'' in homotopy". Definition using holes All definitions below consider a topological space ''X''. A hole in ''X'' is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point., Section 4.3 Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopical Connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of ''n''-connectedness generalizes the concepts of path-connectedness and simple connectedness. An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is ''n''-connected (or ''n''-simple connected) if its first ''n'' homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is ''n''-connected if it is an isomorphism "up to dimension ''n,'' in homotopy". Definition using holes All definitions below consider a topological space ''X''. A hole in ''X'' is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point., Section 4.3 Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurewicz Theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. Statement of the theorems The Hurewicz theorems are a key link between homotopy groups and homology groups. Absolute version For any path-connected space ''X'' and positive integer ''n'' there exists a group homomorphism :h_* \colon \pi_n(X) \to H_n(X), called the Hurewicz homomorphism, from the ''n''-th homotopy group to the ''n''-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator u_n \in H_n(S^n), then a homotopy class of maps f \in \pi_n(X) is taken to f_*(u_n) \in H_n(X). The Hurewicz theorem states cases in which the Hurewitz homomorphism is an isomorphism. * For n\ge 2, if ''X'' is (n-1)-connected (that is: \pi_i(X)= 0 for all ''i''2 there exists a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ÄŒech Complex
In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distribution it is drawn from. Given a finite point cloud ''X'' and an ''ε'' > 0, we construct the Čech complex \check C_\varepsilon(X) as follows: Take the elements of ''X'' as the vertex set of \check C_\varepsilon(X) . Then, for each \sigma\subset X , let \sigma\in \check C_\varepsilon(X) if the set of ''ε''-balls centered at points of σ has a nonempty intersection. In other words, the Čech complex is the nerve of the set of ''ε''-balls centered at points of ''X''. By the nerve lemma, the Čech complex is homotopy equivalent to the union of the balls. Relation to Vietoris–Rips complex The Čech complex is a subcomplex of the Vietoris–Rips complex. While the Čech complex is more computationally expensive than the Vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) where X is a set of elements called ''nodes'' or ''vertices'', and E is a set of non-empty subsets of X called '' hyperedges'' or ''edges''. Therefore, E is a subset of \mathcal(X) \setminus\, where \mathcal(X) is the power set of X. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''. A directed hypergraph differs in that its hyperedges are not sets, but ordered pairs of subsets of X, with each pair's first and second entries constituting the tail and head of the hyperedge respectively. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial from an abstract simplicial complex, the former is often called a geometric simplicial complex.'', Section 4.3'' Definitions A simplicial complex \mathcal is a set of simplices that satisfies the following conditions: :1. Every face of a simplex from \mathcal is also in \mathcal. :2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial ''k''-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence Complex
The independence complex of a graph is a mathematical object describing the independent sets of the graph. Formally, the independence complex of an undirected graph ''G'', denoted by I(''G''), is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the independent sets of ''G''. Any subset of an independent set is itself an independent set, so I(''G'') is indeed closed under taking subsets. Every independent set in a graph is a clique in its complement graph, and vice versa. Therefore, the independence complex of a graph equals the clique complex of its complement graph, and vice versa. Homology groups Several authors studied the relations between the properties of a graph ''G'' = (''V'', ''E''), and the homology groups of its independence complex I(''G''). In particular, several properties related to the dominating sets in ''G'' guarantee that some reduced homology groups of I(''G'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \operatorname B^n or \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Groups
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
