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 ...

, a -chain is a

formal linear combination In mathematics, a formal sum, formal series, or formal linear combination may be: *In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients. *In linear algebra, an ...

of the -cells in a

cell 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 clas ...

. In

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 ...

es (respectively,

cubical complex In mathematics, a cubical complex (also called cubical set and Cartesian complex) is a set composed of points, line segments, squares, cubes, and their ''n''-dimensional counterparts. They are used analogously to simplicial complexes and CW comp ...

es), -chains are combinations of -simplices (respectively, -cubes), but not necessarily connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains.

Definition

For a

X

, the group

C_n(X)

n

-chains of

X

is given by:

C_n(X) = \left\

where

\sigma_i

are singular

n

-simplices of

X

. Note that any element in

C_n(X)

not necessary to be a connected simplicial complex.

Integration on chains

Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients (which are typically integers). The set of all ''k''-chains forms a group and the sequence of these groups is called a

chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...

Boundary operator on chains

The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a ''k''-chain is a (''k''−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator. Example 1: The boundary of a path is the formal difference of its endpoints: it is a

telescoping sum In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums only consists of two terms of (a_n) after c ...

. To illustrate, if the 1-chain

c = t_1 + t_2 + t_3\,

is a path from point

v_1\,

to point

v_4\,

, where

t_1=_1, v_2,

t_2=

_2, v_3 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

, and

t_3=_3, v_4,

are its constituent 1-simplices, then

+ \partial_1(

+ \partial_1( _3, v_4 \\ &= ( _2 _1 + ( _3 _2 + ( _4 _3 \\ &= _4 _1 \end Example 2: The boundary of the triangle is a formal sum of its edges with signs arranged to make the traversal of the boundary counterclockwise. A chain is called a cycle when its boundary is zero. A chain that is the boundary of another chain is called a boundary. Boundaries are cycles, so chains form a

, whose homology groups (cycles modulo boundaries) are called simplicial homology groups. Example 3: The plane punctured at the origin has nontrivial 1-homology group since the unit circle is a cycle, but not a boundary. In

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

, the duality between the boundary operator on chains and the

exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...

is expressed by the general

Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...

References

{{reflist Algebraic topology Integration on manifolds