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
where ai's are integers and Si's are simplexes. The addition of two chains is defined component-wise:
Such addition is naturally commutative and makes the set of chains an additive group.
- 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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