Chain (algebraic topology)
In algebraic topology, a -chain
is a formal linear combination of the -cells in a cell complex. In simplicial complexes, -chains are combinations of -simplices, but not necessarily connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains.
Definition
For a simplicial complex, the group of -chains of is given by:where are singular -simplices of. Note that an element in is not necessarily 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.The set of all k-chains forms a group and the sequence of these groups is called a chain 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 -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. To illustrate, if the 1-chain is a path from point to point, where
and
are its constituent 1-simplices, then
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 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, the duality between the boundary operator on chains and the exterior derivative is expressed by the general Stokes' theorem.