Addition of Chains

Complexes and chains are two fundamental objects of algebraic topology and homological algebra. Complexes consist of simplexes, which are just the most common geometrical entities: points (vertices), finite simple curves (edges), plane regions (faces) bounded by a closed sequence of edges. Vertices are 0-simplexes, edges are 1-simplexes, faces are 2-simplexes. Of course there are generalizations to higher dimensions. A complex is a topological space obained from the union of a family of simplexes, with some simplexes possibly identified. Gluing the edges of a paper rectangle provides a model for such (topological) identification.

(Integral) chains are sefined as the formal sums

a1S1 + a2S2 + ... + anSn,

where ai's are integers and Si's are simplexes. The addition of two chains is defined component-wise:

(a1S1 + ... + anSn) + (b1S1 + ... + bnSn) = (a1 + b1)S1 + ...+ (an + bn)S1.

Such addition is naturally commutative and makes the set of chains an additive group.


References

  1. P. Alexandroff, Elementary Concepts of Topology, Dover Publications, NY, 1961
  2. M. Henle, A Combinatorial Introduction to Topology, Dover Publications, NY, 1979
  3. K. Janich, Topology, UTM, Springer-Verlag, 1984
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