boundary operator

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a chain complex is an
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
that consists of a sequence of
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s (or
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...
) and a sequence of between consecutive groups such that the
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of each homomorphism is included in the
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its
cohomology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. In
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 t ...
, the singular chain complex of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
X is constructed using continuous maps from a
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space. For e ...

to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the
singular homology In 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 ...
of X, and is a commonly used invariant of a topological space. Chain complexes are studied in
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
, but are used in several areas of mathematics, including
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,
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differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast E ...
and
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. They can be defined more generally in
abelian categories In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mot ...
.

Definitions

A chain complex $\left(A_\bullet, d_\bullet\right)$ is a sequence of abelian groups or modules ..., ''A''0, ''A''1, ''A''2, ''A''3, ''A''4, ... connected by homomorphisms (called boundary operators or differentials) , such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy , or with indices suppressed, . The complex may be written out as follows. ::$\cdots \xleftarrow A_0 \xleftarrow A_1 \xleftarrow A_2 \xleftarrow A_3 \xleftarrow A_4 \xleftarrow \cdots$ The cochain complex $\left(A^\bullet, d^\bullet\right)$ is the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
notion to a chain complex. It consists of a sequence of abelian groups or modules ..., ''A''0, ''A''1, ''A''2, ''A''3, ''A''4, ... connected by homomorphisms satisfying . The cochain complex may be written out in a similar fashion to the chain complex. ::$\cdots \xrightarrow A^0 \xrightarrow A^1 \xrightarrow A^2 \xrightarrow A^3 \xrightarrow A^4 \xrightarrow \cdots$ The index ''n'' in either ''A''''n'' or ''A''''n'' is referred to as the degree (or dimension). The difference between chain and cochain complexes is that, in chain complexes, the differentials decrease dimension, whereas in cochain complexes they increase dimension. All the concepts and definitions for chain complexes apply to cochain complexes, except that they will follow this different convention for dimension, and often terms will be given the
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''co-''. In this article, definitions will be given for chain complexes when the distinction is not required. A bounded chain complex is one in which
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the ''A''''n'' are 0; that is, a finite complex extended to the left and right by 0. An example is the chain complex defining the
simplicial homology In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space. For e ...
of a finite
simplicial complex In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. A chain complex is bounded above if all modules above some fixed degree ''N'' are 0, and is bounded below if all modules below some fixed degree are 0. Clearly, a complex is bounded both above and below if and only if the complex is bounded. The elements of the individual groups of a (co)chain complex are called (co)chains. The elements in the kernel of ''d'' are called (co)cycles (or closed elements), and the elements in the image of ''d'' are called (co)boundaries (or exact elements). Right from the definition of the differential, all boundaries are cycles. The ''n''-th (co)homology group ''H''''n'' (''H''''n'') is the group of (co)cycles (co)boundaries in degree ''n'', that is, ::$H_n = \ker d_/\mbox d_ \quad \left\left(H^n = \ker d^/\mbox d^ \right\right)$

Exact sequences

An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A short exact sequence is a bounded exact sequence in which only the groups ''A''''k'', ''A''''k''+1, ''A''''k''+2 may be nonzero. For example, the following chain complex is a short exact sequence. :$\cdots \xrightarrow \; 0 \; \xrightarrow \; \mathbf \; \xrightarrow \; \mathbf \twoheadrightarrow \mathbf/p\mathbf \; \xrightarrow \; 0 \; \xrightarrow \cdots$ In the middle group, the closed elements are the elements pZ; these are clearly the exact elements in this group.

Chain maps

A chain map ''f'' between two chain complexes $\left(A_\bullet, d_\right)$ and $\left(B_\bullet, d_\right)$ is a sequence $f_\bullet$ of homomorphisms $f_n : A_n \rightarrow B_n$ for each ''n'' that commutes with the boundary operators on the two chain complexes, so $d_ \circ f_n = f_ \circ d_$. This is written out in the following
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. : A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology $\left(f_\bullet\right)_*:H_\bullet\left(A_\bullet, d_\right) \rightarrow H_\bullet\left(B_\bullet, d_\right)$. A continuous map ''f'' between topological spaces ''X'' and ''Y'' induces a chain map between the singular chain complexes of ''X'' and ''Y'', and hence induces a map ''f''* between the singular homology of ''X'' and ''Y'' as well. When ''X'' and ''Y'' are both equal to the ''n''-sphere, the map induced on homology defines the
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of the map ''f''. The concept of chain map reduces to the one of boundary through the construction of the Mapping cone (homological algebra), cone of a chain map.

Chain homotopy

A chain homotopy offers a way to relate two chain maps that induce the same map on homology groups, even though the maps may be different. Given two chain complexes ''A'' and ''B'', and two chain maps , a chain homotopy is a sequence of homomorphisms such that . The maps may be written out in a diagram as follows, but this diagram is not commutative. : The map ''hd''''A'' + ''d''''B''''h'' is easily verified to induce the zero map on homology, for any ''h''. It immediately follows that ''f'' and ''g'' induce the same map on homology. One says ''f'' and ''g'' are chain homotopic (or simply homotopic), and this property defines an equivalence relation between chain maps. Let ''X'' and ''Y'' be topological spaces. In the case of singular homology, a homotopy between continuous maps induces a chain homotopy between the chain maps corresponding to ''f'' and ''g''. This shows that two homotopic maps induce the same map on singular homology. The name "chain homotopy" is motivated by this example.

Examples

Singular homology

Let ''X'' be a topological space. Define ''C''''n''(''X'') for Natural number, natural ''n'' to be the free abelian group formally generated by singular homology, singular n-simplices in ''X'', and define the boundary map $\partial_n: C_n\left(X\right) \to C_\left(X\right)$ to be ::$\partial_n : \, \left(\sigma: \left[v_0,\ldots,v_n\right] \to X\right) \mapsto \left(\sum_^n \left(-1\right)^i \sigma: \left[v_0,\ldots, \hat v_i, \ldots, v_n\right] \to X\right)$ where the hat denotes the omission of a vertex (geometry), vertex. That is, the boundary of a singular simplex is the alternating sum of restrictions to its faces. It can be shown that ∂2 = 0, so $\left(C_\bullet, \partial_\bullet\right)$ is a chain complex; the singular homology $H_\bullet\left(X\right)$ is the homology of this complex. Singular homology is a useful invariant of topological spaces up to homotopy#homotopy equivalence, homotopy equivalence. The degree zero homology group is a free abelian group on the connected space#Path connectedness, path-components of ''X''.

de Rham cohomology

The differential form, differential ''k''-forms on any smooth manifold ''M'' form a real number, real vector space called Ω''k''(''M'') under addition. The exterior derivative ''d'' maps Ω''k''(''M'') to Ω''k''+1(''M''), and ''d'' = 0 follows essentially from symmetry of second derivatives, so the vector spaces of ''k''-forms along with the exterior derivative are a cochain complex. :$\Omega^0\left(M\right)\ \stackrel\ \Omega^1\left(M\right) \to \Omega^2\left(M\right) \to \Omega^3\left(M\right) \to \cdots$ The cohomology of this complex is called the de Rham cohomology of ''X''. The homology group in dimension zero is isomorphic to the vector space of locally constant functions from ''M'' to R. Thus for a compact manifold, this is the real vector space whose dimension is the number of connected components of ''M''. smoothness#smooth functions between manifolds, Smooth maps between manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies.

Category of chain complexes

Chain complexes of ''K''-modules with chain maps form a category (mathematics), category Ch''K'', where ''K'' is a commutative ring. If ''V'' = ''V''$_*$ and ''W'' = ''W''$_*$ are chain complexes, their tensor product $V \otimes W$ is a chain complex with degree ''n'' elements given by :$\left(V \otimes W\right)_n = \bigoplus_ V_i \otimes W_j$ and differential given by : $\partial \left(a \otimes b\right) = \partial a \otimes b + \left(-1\right)^ a \otimes \partial b$ where ''a'' and ''b'' are any two homogeneous vectors in ''V'' and ''W'' respectively, and $\left, a\$ denotes the degree of ''a''. This tensor product makes the category Ch''K'' into a symmetric monoidal category. The identity object with respect to this monoidal product is the base ring ''K'' viewed as a chain complex in degree 0. The braided monoidal category, braiding is given on simple tensors of homogeneous elements by :$a \otimes b \mapsto \left(-1\right)^ b \otimes a$ The sign is necessary for the braiding to be a chain map. Moreover, the category of chain complexes of ''K''-modules also has closed monoidal category, internal Hom: given chain complexes ''V'' and ''W'', the internal Hom of ''V'' and ''W'', denoted Hom(''V'',''W''), is the chain complex with degree ''n'' elements given by $\Pi_\text_K \left(V_i,W_\right)$ and differential given by : $\left(\partial f\right)\left(v\right) = \partial\left(f\left(v\right)\right) - \left(-1\right)^ f\left(\partial\left(v\right)\right)$. We have a natural isomorphism :$\text\left(A\otimes B, C\right) \cong \text\left(A,\text\left(B,C\right)\right)$

Further examples

*Amitsur complex *A complex used to define Bloch's higher Chow groups *Buchsbaum–Rim complex *Čech complex *Cousin complex *Eagon–Northcott complex *Gersten complex *Graph complexhttps://ncatlab.org/nlab/show/graph+complex *Koszul complex *Moore complex *Schur complex