homology

Etymology
From.

In topology, first used by French polymath, in the sense (close to what is now called a ) 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 , which is what mathematicians now mean by homology.

Noun

 * 1) The relationship of being homologous; a homologous relationship.
 * 2)  specifically, such relationship in the context of the geometry of perspective.
 * 3) * 1863,, A Treatise on Conic Sections, , 4th Edition, page 61,
 * Two triangles are said to be homologous, when the intersections of the corresponding sides lie on the same right line called the axis of homology: prove that the lines joining the corresponding vertices meet in a point [called the centre of homology].
 * 1) * 1885, Charles Leudesdorf (translator),, Elements of Projective Geometry, (Clarendon Press), page 11,
 * Two corresponding straight lines therefore always intersect on a fixed straight line, which we may call s; thus the given figures are in homology, O being the centre, and s the axis, of homology.
 * 1)  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.
 * 2)  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.
 * 3)  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.
 * 4)  The relationship, between elements, of being in the same group of the periodic table.
 * 5)  The relationship, between organic compounds, of being in the same homologous series.
 * 6)   The relationship, between characteristics or behaviours, of having a shared evolutionary or developmental origin;  specifically, a correspondence between structures in separate life forms having a common evolutionary origin, such as that between mammalian flippers and hands.
 * 7) * 2000, Julie A. Hawkins, Chapter 2: A survey of primary homology assessment, Robert Scotland, R. Toby Pennington (editors), Homology and Systematics, Taylor & Francis,, page 22,
 * The objective of this study is to classify approaches to primary homology assessment, and to quantify the extent to which different approaches are found in the literature by examining variation in the ways characters are defined and coded in a data matrix.
 * 1)  The presence of the same series of bases in different but related genes.
 * 2)  The relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.
 * 1) * 2000, Julie A. Hawkins, Chapter 2: A survey of primary homology assessment, Robert Scotland, R. Toby Pennington (editors), Homology and Systematics, Taylor & Francis,, page 22,
 * The objective of this study is to classify approaches to primary homology assessment, and to quantify the extent to which different approaches are found in the literature by examining variation in the ways characters are defined and coded in a data matrix.
 * 1)  The presence of the same series of bases in different but related genes.
 * 2)  The relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.
 * 1)  The relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.

Usage notes

 * When used attributively with the name of a topological space (such as in the terms homology n-sphere and homology manifold) the reference is to a space whose homology is the same as that of the named space: thus, for example, a homology manifold is a space whose homology is that of some manifold.
 * Sometimes used to mean ': thus, X did Y by computing the homology' of Z means X did Y by computing the homology groups of Z''.
 * More loosely, the term homology in a space refers to a (group of singular homologies).
 * For a discussion of the use of the term (and ) in biology, see: 1998 Nov,, "Homology in Classical and Molecular Biology", ', 5''', No. 6: 603–625, (accessed 18 Dec 2009; 18 Dec 2009).
 * More loosely, the term homology in a space refers to a (group of singular homologies).
 * For a discussion of the use of the term (and ) in biology, see: 1998 Nov,, "Homology in Classical and Molecular Biology", ', 5''', No. 6: 603–625, (accessed 18 Dec 2009; 18 Dec 2009).
 * For a discussion of the use of the term (and ) in biology, see: 1998 Nov,, "Homology in Classical and Molecular Biology", ', 5''', No. 6: 603–625, (accessed 18 Dec 2009; 18 Dec 2009).

Translations

 * Finnish: ,
 * Irish: homalógacht
 * Portuguese: homologia
 * Romanian:


 * Finnish: ,
 * Irish: homalógacht
 * Portuguese: homologia
 * Romanian:


 * Chinese:
 * Mandarin:, 同调论
 * Finnish:
 * Irish: homalógacht
 * Portuguese: homologia


 * Chinese:
 * Mandarin:, 同调群
 * Finnish:
 * Irish: homalógacht
 * Portuguese: homologia


 * Finnish:
 * Irish: homalógacht
 * Romanian:


 * Finnish:
 * French:
 * Irish: homalógacht
 * Italian:
 * Portuguese: homologia
 * Spanish: homología


 * Finnish:
 * French:
 * Irish: homalógacht
 * Italian:
 * Portuguese: homologia
 * Spanish: homología


 * Finnish:
 * French:
 * Irish: homalógacht
 * Italian:
 * Portuguese: homologia
 * Spanish: homología


 * Czech:
 * French:
 * German:
 * Romanian:
 * Slovene: