Chain complexes arise in abundance in algebra and algebraic topology. For example, if "X" is a topological space then the singular chains "C""n"("X") are formal linear combinations of continuous maps from the standard "n"-simplex into "X"; if "K" is a simplicial complex then the simplicial chains "C""n"("K") are formal linear combinations of the "n"-simplices of "K"; if "A" = "F"/"R" is a presentation of an abelian group "A" by generators and relations, where "F" is a free abelian group spanned by the generators and "R" is the subgroup of relations, then letting "C"1("A") = "R", "C"0("A") = "F", and "C""n"("A") = 0 for all other "n" defines a sequence of abelian groups. In all these cases, there are natural differentials "d""n" making "C""n" into a chain complex, whose homology reflects the structure of the topological space "X", the simplicial complex "K", or the abelian group "A". In the case of topological spaces, we arrive at the notion of singular homology, which plays a fundamental role in investigating the properties of such spaces, for example, manifolds.