homology

Meanings

1. The relationship of being homologous; a homologous relationship.
2. specifically, such relationship in the context of the geometry of perspective.
3. An automorphism of the projective plane (representing a perspective projection) that leaves all the points of some straight line (the homology axis) fixed and maps all the lines through some single point (the homology centre) onto themselves.
4. A general way of associating a sequence of algebraic objects, such as abelian groups or modules, to a sequence of topological spaces; also used attributively: see Usage notes below.
5. Given a chain complex {Gₙ} and its associated set of homomorphisms {Hₙ}, the rule which explains how each Hₙ maps Gₙ into the kernel of Gₙ₊₁.
6. The relationship, between elements, of being in the same group of the periodic table.
7. The relationship, between organic compounds, of being in the same homologous series.
8. The relationship, between characteristics or behaviours, of having a shared evolutionary or developmental origin;
9. The relationship, between characteristics or behaviours, of having a shared evolutionary or developmental origin; (evolutionary theory) specifically, a correspondence between structures in separate life forms having a common evolutionary origin, such as that between mammalian flippers and hands.
10. specifically, a correspondence between structures in separate life forms having a common evolutionary origin, such as that between mammalian flippers and hands.
11. The presence of the same series of bases in different but related genes.
12. The relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.

Etymology

From Latin homologia, from Ancient Greek ὁμολογία (homología, “agreement, assent”); compare French homologie. By surface analysis, homo- + -logy. In topology, first used by French polymath Henri Poincaré, in the sense (close to what is now called a bordism) of a relation between manifolds mapped into a reference manifold: that is, the property of such manifolds that they form the boundary of a higher-dimensional manifold inside the reference manifold. Poincaré's version was eventually replaced by the more general singular homology, which is what mathematicians now mean by homology.

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