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:
- 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