homology

noun

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.

1. The relationship of being homologous; a homologous relationship.

   “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. The relationship of being homologous; a homologous relationship.

   “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. The relationship of being homologous; a homologous relationship.

   “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.”

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

4. The relationship of being homologous; a homologous relationship.

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

5. The relationship of being homologous; a homologous relationship.
6. The relationship of being homologous; a homologous relationship.
7. The relationship of being homologous; a homologous relationship.

   “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.”

8. The relationship of being homologous; a homologous relationship.
9. The relationship of being homologous; a homologous relationship.
10. The relationship of being homologous; a homologous relationship.