This article is about algebraic topology. For the term chain in order theory see chain (order theory) Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

In algebraic topology, a simplicial k-chain is a formal linear combination of k-simplices. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... simplex refers to a one-way communications channel. ...

Integration on chains

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

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.

Example 1: The boundary of a directed path is the formal difference of its endpoints.

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 chain complex, whose homology groups (cycles modulo boundaries) are called simplicial homology groups. In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ... 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). ...

Example 3: A 0-cycle is a linear combination of points such that the sum of all the coefficients is 0. Thus, the 0-homology group measures the number of path connected components of the space.

Example 4: 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, the duality between the boundary operator on chains and the exterior derivative is expressed by the general Stokes' theorem. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... The Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...