In mathematics, the term was first used in a topological sense by French polymath Henri Poincaré, in a sense close to what is now called a bordism. Thus, he thought of a homology as a relation between manifolds mapped into a reference manifold. Such manifolds form a homology when they form the boundary of a higher-dimensional manifold inside the reference manifold. Poincaré simplified his definition by considering only spaces that were simplicial complexes (that had a triangulation), and only objects in the space made up of simplices in the triangulation. Eventually, Poincaré's version was replaced by the more general singular homology, which is what mathematicians mean by homology.
In modern usage, the term homology often means homology group. Thus, X did Y by computing the homology of Z means X did Y by computing the homology groups of Z. More loosely, as used in homology in a space, the term corresponds to a singular homology group (group of singular homologies).
homology (plural homologies)
- A homologous relationship.
- (biology) A correspondence of structures in two life forms with a common evolutionary origin, such as flippers and hands.
- (chemistry) The relationship between the elements in the same group of the periodic table, or between organic compounds in a homologous series.
- (topology) A theory associating a system of groups to each topological space.
- (algebra) A certain system of groups associated to a chain complex.
- (genetics) The presence of the same series of bases in related genes.
- Like many terms that start with a non-silent h but have emphasis on their second syllable, some people precede homology with an, others with a.
- (biology): For a discussion of the use of the term homology (and by association homologous) in biology, see: Patterson, Colin. "Homology in Classical and Molecular Biology." Molecular Biology and Evolution 5, no. 6 (November 1988): 603–625. http://mbe.oxfordjournals.org/cgi/reprint/5/6/603.pdf (accessed 18 December 2009; archived 18 December 2009, http://www.webcitation.org/5m7rn4rCe )
- (mathematics): When used attributively before the name of a topological space, as in homology n-sphere or homology manifold, the reference is to a space whose homology is the same as that of the space mentioned: an homology manifold, for example, is a space whose homology is that of some manifold.
- (mathematics): In modern usage, the term is used to mean homology group. Thus, X did Y by computing the homology of Z means X did Y by computing the homology groups of Z. More loosely, as used in homology in a space, the term corresponds to a singular homology group (group of singular homologies).
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- Eilenberg–Steenrod axioms on Wikipedia.Wikipedia:Eilenberg–Steenrod axioms
- Singular homology on Wikipedia.Wikipedia:Singular homology