In mathematics, a chain complex is a construct originally used in the field of algebraic topology. 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. More generally, homological algebra includes the study of chain complexes in the abstract, without any reference to an underlying space. In this case, chain complexes are studied axiomatically as algebraic structures. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ...

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). Chain complexes are easily defined in abelian categories, also. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...

Formal definition

A chain complex is a sequence of abelian groups or modules 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. They tend to be written out like so: In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In abstract algebra, a homomorphism is a structure-preserving map. ...

A variant on the concept of chain complex is that of cochain complex. A cochain complex is a sequence of abelian groups or modules A₀, A₁, A₂, ... 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: In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In abstract algebra, a homomorphism is a structure-preserving map. ...

The idea is basically the same. 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.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. 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. In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

Fundamental terminology

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

d² = 0.

The image of d is the group of boundaries, or in a cochain complex, coboundaries. 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 homology groups. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

Examples

Singular homology

Suppose we are given a topological space X. In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... Topological spaces are structures that allow one to formalize 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 In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... 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 one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...

where the hat denotes the omission of a vertex. That is, the boundary of a singular simplex is alternating sum of restrictions to its faces. It can be shown ∂² = 0, so is a chain complex; the singular homology is the homology of this complex; that is, In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...

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. The exterior derivative d = d _k maps Ω^k(M) → 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: In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... 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 (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a function f(x1, x2, ..., xn) of n variables. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ...

The homology of this complex is the de Rham cohomology

Chain maps

A chain map f between two chain complexes and is a collection of maps for each n that intertwines with the differentials on the two chain complexes: . Such a map sends cycles to cycles and boundaries to boundaries, and thus descends to a map on homology:.

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. 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 mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... Did somebody just say functor? In category theory, 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. The notion is modelled on the homotopy concept for mappings of topological space, and reflects the fact that homotopic maps of continuous spaces induce the same maps on their homology. Chain homotopies have a geometric interpretation; it is described, for example, in the book of Bott and Tu. The two bold paths shown above are homotopic relative to their endpoints. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

Let (A_n, d_n) and (B_n, d′_n) be chain complexes and f, g be chain maps from the first to the second. A chain homotopy between f and g is given by a homotopy operator, a sequence of homomorphisms D_n from A_n to B_n+1 such that

f − g = Dd + d′D,

or adorned with full indices, as can be easily reconstructed for diagram chasing, In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

f_n − g_n = D_n−1d_n + d′_n+1D_n.

The chain maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology.

A weaker notion of equivalence of maps between chain complexes is that of a quasi-isomorphism, which is a chain map in which the induced map on homology is an isomorphism. The derived category has chain complexes as objects, but morphisms are built by inverting quasi-isomorphisms. In homological algebra, a branch of mathematics, a quasi-isomorphism is a morphism A → B of chain or cochain complexes such that the induced morphisms Hn(A•, Z) → Hn(B•, Z) of homology groups or Hn(A•, Z) → Hn(B•, Z) of cohomology groups are isomorphisms for all n ≥ 0. ... In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ...

See also

• Homology
• Differential graded algebra

In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure. ...

References

• Raoul Bott and Loring Tu, Differential Forms in Algebraic Topology. Springer-Verlag, 1982.