homology

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]

homology (countable and uncountable, plural homologies)

1. A homologous relationship.
2. (topology) A theory associating a system of groups to each topological space.
3. (algebra) A certain system of groups associated to a chain complex.
4. (chemistry) The relationship between the elements in the same group of the periodic table, or between organic compounds in a homologous series.
5. (evolutionary theory) A correspondence of structures in two life forms with a common evolutionary origin, such as flippers and hands.
6. (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.
• (evolutionary theory):
  • For a discussion of the use of the term homology (and 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 )
• (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]

• cohomology
• microhomology
• semihomology

• analogy
• homologue

• Eilenberg–Steenrod axioms on Wikipedia.Wikipedia
• Singular homology on Wikipedia.Wikipedia