|
Exact Couple
In mathematics, an exact couple, due to , is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple. For the definition of an exact couple and the construction of a spectral sequence from it (which is immediate), see . For a basic example, see Bockstein spectral sequence. The present article covers additional materials. Exact couple of a filtered complex Let ''R'' be a ring, which is fixed throughout the discussion. Note if ''R'' is \Z, then modules over ''R'' are the same thing as abelian groups. Each filtered chain complex of modules determines an exact couple, which in turn determines a spectral sequence, as follows. Let ''C'' be a chain complex graded by integers and suppose it is given an increasing filtration: for each integer ''p'', there is an inclusion of complexes: :F_ C \subset F_p C. From the filtration one can form the associated graded comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf (mathematics), sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the direct image of a sheaf, pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomolo ... [...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 invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, 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 gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre Spectral Sequence
In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space ''X'' of a (Serre) fibration in terms of the (co)homology of the base space ''B'' and the fiber ''F''. The result is due to Jean-Pierre Serre in his doctoral dissertation. Cohomology spectral sequence Let f\colon X\to B be a Serre fibration of topological spaces, and let ''F'' be the (path-connected) fiber. The Serre cohomology spectral sequence is the following: : E_2^ = H^p(B, H^q(F)) \Rightarrow H^(X). Here, at least under standard simplifying conditions, the coefficient group in the E_2-term is the ''q''-th integral cohomology group of ''F'', and the outer group is the singular cohomology of ''B'' with coefficients in that group. Strictly speaking, what is meant is cohomology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bockstein Spectral Sequence
In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod ''p'' coefficients and the homology reduced mod ''p''. It is named after Meyer Bockstein. Definition Let ''C'' be a chain complex of torsion-free abelian groups and ''p'' a prime number. Then we have the exact sequence: :0 \longrightarrow C \overset\longrightarrow C \overset \longrightarrow C \otimes \Z/p \longrightarrow 0. Taking integral homology ''H'', we get the exact couple of "doubly graded" abelian groups: :H_*(C) \overset \longrightarrow H_*(C) \overset \longrightarrow H_*(C \otimes \Z/p) \overset \longrightarrow. where the grading goes: H_*(C)_ = H_(C) and the same for H_*(C \otimes \Z/p),\deg i = (1, -1), \deg j = (0, 0), \deg k = (-1, 0). This gives the first page of the spectral sequence: we take E_^1 = H_(C \otimes \Z/p) with the differential ^1 d = j \circ k. The derived couple of the above exact couple then gives the second page and so forth. Explicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Graded Module
In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring: :\operatorname_I R = \oplus_^\infty I^n/I^. Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the graded module over \operatorname_I R: :\operatorname_I M = \oplus_^\infty I^n M/ I^ M. Basic definitions and properties For a ring ''R'' and ideal ''I'', multiplication in \operatorname_IR is defined as follows: First, consider homogeneous elements a \in I^i/I^ and b \in I^j/I^ and suppose a' \in I^i is a representative of ''a'' and b' \in I^j is a representative of ''b''. Then define ab to be the equivalence class of a'b' in I^/I^. Note that this is well-defined modulo I^. Multiplication of inhomogeneous elements is defined by using the distributive property. A ring or module may be related to its associated graded ring or module through the initial form map. Let ''M'' be an ''R''-module and ''I'' an ideal of ''R''. Given f \in M, the ini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf (mathematics), sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the direct image of a sheaf, pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. Formal definitions Homotopy lifting property A mapping p \colon E \to B satisfies the homotopy lifting property for a space X if: * for every homotopy h \colon X \times , 1\to B and * for every mapping (also called lift) \tilde h_0 \colon X \to E lifting h, _ = h_0 (i.e. h_0 = p \circ \tilde h_0) there exists a (not necessarily unique) homotopy \tilde h \colon X \times , 1\to E lifting h (i.e. h = p \circ \tilde h) with \tilde h_0 = \tilde h, _. The following commutative diagram shows the situation:^ Fibration A fibration (also called Hurewicz fibration) is a mapping p \colon E \to B satisfying the homotopy lifting property for all spaces X. The space B is called base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CW Complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces\emptyset = X_ \subset X_0 \subset X_1 \subset \cdotssuch that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to D^k, to X_ by continuous gluing maps g^k_\alpha: \partial e^k_\alpha \to X_. The maps are also called attaching maps. Each X_k is called the k-skeleton of the complex. The topology of X = \cup_ X_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is ''locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected Space
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 (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. 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 circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Coefficient System
In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. Definition Let ''X'' be a topological space. A local system (of abelian groups/modules/...) on ''X'' is a locally constant sheaf (of abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...s/Module over a ring, modules...) on ''X''. In other words, a sheaf \mathcal is a local system if every point has an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
