# 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

_{1}S

_{1}+ a

_{2}S

_{2}+ ... + a

_{n}S

_{n},

where a_{i}'s are integers and S_{i}'s are simplexes. The addition of two chains is defined component-wise:

_{1}S

_{1}+ ... + a

_{n}S

_{n}) + (b

_{1}S

_{1}+ ... + b

_{n}S

_{n}) = (a

_{1}+ b

_{1})S

_{1}+ ...+ (a

_{n}+ b

_{n})S

_{1}.

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

## References

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

### What Can Be Added?

- What Is Addition?
- Addition of Chains
- Addition of Equations
- Addition of Functions
- Addition of Numbers
- Addition of Sets
- Addition of Shapes
- Addition of Spaces
- Addition of Strings
- Addition of Vectors

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