Homology (mathematics)

Lore & Background

In mathematics, the term homology has three primary, closely related usages. First, there is the homology of a chain complex, a sequence of abelian groups called homology groups, which are regarded as fundamental invariants of the chain complex. The nth homology group is defined as the quotient group of cycles modulo boundaries. Second, when one can associate a chain complex to a different mathematical object, one can also associate its homology to that object; distinct procedures of associating chain complexes to a given object are grouped into homology theories. Third, homology is important in the study of topological spaces; under nice conditions in which distinct homology theories for a single topological space produce the same homology groups, one can define a single homology of a topological space.

Reader's Guide

Homology is a central concept in algebraic topology and homological algebra, providing invariants for chain complexes, mathematical objects, and topological spaces. The homology of a chain complex is defined via cycles and boundaries, yielding quotient groups that capture structural information. Homology theories, such as singular homology, Morse homology, Khovanov homology, and Hochschild homology, are derived by prescribing chain complexes to mathematical objects and ensuring consistent homology groups. In the language of category theory, a homology theory is a type of functor, and can be formulated as derived functors measuring the failure of exactness. For topological spaces, under the Eilenberg–Steenrod axioms, different homology theories yield the same groups as singular homology for sufficiently nice spaces. Homology helps distinguish mathematical objects and provides insight into their structure, with applications ranging from graph homology to cohomology theories.

Did You Know?

• The nth homology group of a chain complex is the quotient group of cycles modulo boundaries.
• For sufficiently nice topological spaces, any homology theory satisfying the Eilenberg–Steenrod axioms yields the same homology groups as singular homology.
• There is a related notion of cohomology of a cochain complex, giving rise to various cohomology theories.