Chain complex - Citizendia

In mathematics, a chain complex is a construct originally used in the field of algebraic topology. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic It is an algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or an algebraic construction such as a simplicial complex. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments More generally, homological algebra includes the study of chain complexes in the abstract, without any reference to an underlying space. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting In this case, chain complexes are studied axiomatically as algebraic structures. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations,

Applications of chain complexes usually define and apply their homology groups (cohomology groups for cochain complexes); in more abstract settings various equivalence relations are applied to complexes (for example starting with the chain homotopy idea). In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain Chain complexes are easily defined in abelian categories, also. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist

1 Formal definition
2 Fundamental terminology
3 Examples
- 3.1 Singular homology
- 3.2 de Rham cohomology
4 Chain maps
5 Chain homotopy
6 See also
7 References

Formal definition

A chain complex $(A_\bullet, d_\bullet)$ is a sequence of abelian groups or modules . An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars . . A_-2, A_-1, A₀, A₁, A₂, . . . connected by homomorphisms (called boundary operators) d_n : A_n→A_n−1, such that the composition of any two consecutive maps is zero: d_n o d_n+1 = 0 for all n. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector They are usually written out as:

$\cdots \to A_{n+1} \begin{matrix} d_{n+1} \\ \to \\ \, \end{matrix} A_n \begin{matrix} d_n \\ \to \\ \, \end{matrix} A_{n-1} \begin{matrix} d_{n-1} \\ \to \\ \, \end{matrix} A_{n-2} \to \cdots \to A_2 \begin{matrix} d_2 \\ \to \\ \, \end{matrix} A_1 \begin{matrix} d_1 \\ \to \\ \, \end{matrix} A_0 \begin{matrix} d_0 \\ \to \\ \, \end{matrix} A_{-1} \begin{matrix} d_{-1} \\ \to \\ \, \end{matrix} A_{-2} \begin{matrix} d_{-2} \\ \to \\ \, \end{matrix} \cdots$

A variant on the concept of chain complex is that of cochain complex. A cochain complex $(A^\bullet, d^\bullet)$ is a sequence of abelian groups or modules A_-2, A_-1, A₀, A₁, A₂, . An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars . . connected by homomorphisms d_n : A_n→A_n+1, such that the composition of any two consecutive maps is zero: d_n+1 o d_n = 0 for all n:

$\cdots \to A_{-2} \begin{matrix} d_{-2} \\ \to \\ \, \end{matrix} A_{-1} \begin{matrix} d_{-1} \\ \to \\ \, \end{matrix} A_0 \begin{matrix} d_0 \\ \to \\ \, \end{matrix} A_1 \begin{matrix} d_1 \\ \to \\ \, \end{matrix} A_2 \to \cdots \to A_{n-1} \begin{matrix} d_{n-1} \\ \to \\ \, \end{matrix} A_n \begin{matrix} d_n \\ \to \\ \, \end{matrix} A_{n+1} \to \cdots.$

The idea is basically the same. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In either case, the index i in A_i is referred to as the degree.

A bounded chain complex is one in which almost all the A_i are 0; i. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses e. , a finite complex extended to the left and right by 0's. An example is the complex defining the homology theory of a (finite) simplicial complex. In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments A chain complex is bounded above if all degrees above some fixed degree N are 0, and is bounded below if all degrees below some fixed degree are 0. Clearly, a complex is bounded above and below iff the complex is bounded. ↔

Fundamental terminology

Leaving out the indices, the basic relation on d can be thought of as

d² = 0.

The elements of the individual groups of a chain complex are called chains (or cochains in the case of a cochain complex. ) The image of d is the group of boundaries, or in a cochain complex, coboundaries. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage The kernel of d (i. In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that e. , the subgroup sent to 0 by d) is the group of cycles, or in the case of a cochain complex, cocycles. From the basic relation, the (co)boundaries lie inside the (co)cycles. This phenomenon is studied in a systematic way using (co)homology groups. In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is

Examples

Singular homology

Suppose we are given a topological space X. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

Define C_n(X) for natural n to be the free abelian group formally generated by singular n-simplices in X, and define the boundary map

$\partial_n: C_n(X) \to C_{n-1}(X): \, (\sigma: [v_0,\ldots,v_n] \to X) \mapsto (\partial_n \sigma = \sum_{i=0}^n (-1)^i \sigma|[v_0,\ldots, \hat v_i, \ldots, v_n]),$

where the hat denotes the omission of a vertex. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Geometry, a vertex (plural "vertices" is a special kind of point. That is, the boundary of a singular simplex is alternating sum of restrictions to its faces. It can be shown ∂² = 0, so $(C_\bullet, \partial_\bullet)$ is a chain complex; the singular homology $H_\bullet(X)$ is the homology of this complex; that is,

$H_n(X) = \ker \partial_n / \mbox{im } \partial_{n+1}.$

de Rham cohomology

The differential k-forms on any smooth manifold M form an abelian group (in fact an R-vector space) called Ω^k(M) under addition. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Addition is the mathematical process of putting things together The exterior derivative d_k maps Ω^k(M) to Ω^k+1(M), and d ² = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex:

$\Omega^0(M)\ \stackrel{d_0}{\to}\ \Omega^1(M) \to \Omega^2(M) \to \Omega^3(M) \to \cdots.$

The homology of this complex is the de Rham cohomology

$H^0_{\mathrm{DR}}(M, F) = \ker d_0 =$ {locally constant functions on M with values in F} $\cong F$ ^{#{connected pieces of M}}

$H^k_{\mathrm{DR}}(M) = \ker d_k / \mathrm{im} \, d_{k-1}.$

Chain maps

A chain map f between two chain complexes $(A_\bullet, d_{A,\bullet})$ and $(B_\bullet, d_{B,\bullet})$ is a collection of module homomorphisms $f_n : A_n \rightarrow B_n$ for each n that intertwines with the differentials on the two chain complexes: $d_{B,n} \circ f_n = f_{n-1} \circ d_{A,n}$ . In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking Partial derivatives of a function In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. In Mathematics, a constant function is a function whose values do not vary and thus are Constant. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Such a map sends cycles to cycles and boundaries to boundaries, and thus descends to a map on homology: $(f_n)_*:H_\bullet(A_\bullet, d_{A,\bullet}) \rightarrow H_\bullet(B_\bullet, d_{B,\bullet})$ .

A continuous map of topological spaces induces chain maps in both the singular and de Rham chain complexes described above (and in general for the chain complex defining any homology theory of topological spaces) and thus a continuous map induces a map on homology. In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly Because the map induced on a composition of maps is the composition of the induced maps, these homology theories are functors from the category of topological spaces with continuous maps to the category of abelian groups with group homomorphisms. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

Chain homotopy

Chain homotopies give an important equivalence relation between chain maps. Chain homotopic chain maps induce the same maps on homology groups. A particular case is that homotopic maps between two spaces X and Y induce the same maps from homology of X to homology of Y. Chain homotopies have a geometric interpretation; it is described, for example, in the book of Bott and Tu. See Homotopy category of chain complexes for further information. In Homological algebra in Mathematics, the homotopy category K(A of chain complexes in an Additive category A is a framework for

References

Bott, Raoul & Tu, Loring W. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, in particular Abstract algebra and Topology, a differential graded algebra is a Graded algebra with an added chain complex structure Raoul Bott, FRS (born September 24 1923, died December 20 2005) was a (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3

Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic

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