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Chain (algebraic topology)

In algebraic topology, a k-chain is a formal linear combination of the k-cells in a cell complex. In simplicial complexes (respectively, cubical complexes), k-chains are combinations of k-simplices (respectively, k-cubes),[1][2][3] but not necessarily connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains.

Definition

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For a simplicial complex , the group of -chains of is given by:

where are singular -simplices of . that any element in not necessary to be a connected simplicial complex.

Integration on chains

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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.

Boundary operator on chains

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The boundary of a polygonal curve is a linear combination of its nodes; in this case, some linear combination of A1 through A6. Assuming the segments all are oriented left-to-right (in increasing order from Ak to Ak+1), the boundary is A6 − A1.
A closed polygonal curve, assuming consistent orientation, has null boundary.

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. 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 (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, the duality between the boundary operator on chains and the exterior derivative is expressed by the general Stokes' theorem.

References

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  1. ^ Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.
  2. ^ Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer. ISBN 978-1441979391. OCLC 697506452.
  3. ^ Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004). Computational homology. Applied Mathematical Sciences. Vol. 157. New York: Springer-Verlag. doi:10.1007/b97315. ISBN 0-387-40853-3. MR 2028588.
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Chain (algebraic topology)
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