Homology of Complexes

Abstract

Here we first meet the basic notions of homology in simple geometric cases where the homology group arises from a boundary operator. In general, an abelian group with a boundary operator is called a “differential group” or, when provided with dimensions, a “chain complex”. This chapter considers the algebraic process of constructing homology and cohomology groups from chain complexes. Basic is the fact (§ 4) that a short exact sequence of complexes gives a long exact sequence of homology groups. As illustrative background, the last sections provide a brief description of the singular homology groups of a topological space.