homology

 

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Homology

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From 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.^[1]

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homology (countable and uncountable, plural homologies)

1. The relationship of being homologous; a homologous relationship.
   1. (geometry, projective geometry) specifically, such relationship in the context of the geometry of perspective.
      • 1863, George Salmon, A Treatise on Conic Sections, Longman, Brown, Green, Longman, and Roberts, 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].

      • 1885, Charles Leudesdorf (translator), Luigi Cremona, Elements of Projective Geometry, Oxford University Press (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.

   2. (geometry, projective geometry) 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]

      If the homology centre lies on the homology axis, the homology is said to be singular or parabolic; otherwise, it is called non-singular or hyperbolic.

   3. (topology, algebraic topology) 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.
      • 2000, Sibe Mardešić, Strong Shape and Homology, Springer, page v:

        One encounters a similar situation in homology theory. Beside singular homology, which is a homotopy invariant, and Čech homology, which is a shape invariant, there exists strong homology, which is a strong shape invariant. In the special case of metric compacta, this homology was introduced by N.E. Steenrod in 1940 and is often referred to as the Steenrod homology.

      • 2002, Nikolai Saveliev, Invariants of Homology 3-Spheres, Springer, page 2:

        Brieskorn homology spheres are a special case of Siefert fibered homology spheres.

   4. (algebra) Given a chain complex {G_n} and its associated set of homomorphisms {H_n}, the rule which explains how each H_n maps G_n into the kernel of G_n+1.

      Because of their connection with both homology and cohomology, chain complexes are an important topic of study in homological algebra.

   5. (chemistry) The relationship, between elements, of being in the same group of the periodic table.
   6. (organic chemistry) The relationship, between organic compounds, of being in the same homologous series.
   8. (biology, psychology) 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.

      Coordinate term: homomorphism

      • 2000, Julie A. Hawkins, Chapter 2: A survey of primary homology assessment, Robert Scotland, R. Toby Pennington (editors), Homology and Systematics, Taylor & Francis, The Systematics Association, 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.

   9. (genetics) The presence of the same series of bases in different but related genes.
   10. (anthropology) The relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.

• 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.
• (topology):
  • 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 homology group: thus, X did Y by computing the homology of Z means X did Y by computing the homology groups of Z.^[1]
  • More loosely, the term homology in a space refers to a singular homology group (group of singular homologies).^[1]
• (evolutionary theory):
  • For a discussion of the use of the term homology (and homologous) in biology, see: 1998 Nov, Colin Patterson, "Homology in Classical and Molecular Biology", Molecular Biology and Evolution, 5, No. 6: 603–625, (accessed 18 Dec 2009; 18 Dec 2009).

• analogy
• homolog, homologue
• homologation
• homological

• homothety
• homotopy

• (mathematics):
•   Perspective (geometry) on Wikipedia.Wikipedia
•   Eilenberg–Steenrod axioms on Wikipedia.Wikipedia
•   Floer homology on Wikipedia.Wikipedia
•   Khovanov homology on Wikipedia.Wikipedia
•   Singular homology on Wikipedia.Wikipedia
•   Homological algebra on Wikipedia.Wikipedia
•   Homological dimension on Wikipedia.Wikipedia
• (other):
•   Homology modeling on Wikipedia.Wikipedia
•   Sequence homology on Wikipedia.Wikipedia