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.


  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

What Can Be Added?

  1. What Is Addition?
  2. Addition of Chains
  3. Addition of Equations
  4. Addition of Functions
  5. Addition of Numbers
  6. Addition of Sets
  7. Addition of Shapes
  8. Addition of Spaces
  9. Addition of Strings
  10. Addition of Vectors

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