Homology

homology, in mathematics, a basic notion of algebraic topology. Intuitively, two curves in a plane or other two-dimensional surface are homologous if together they bound a region—thereby distinguishing between an inside and an outside. Similarly, two surfaces within a three-dimensional space are homologous if together they bound a three-dimensional region lying within the ambient space.

There are many ways of making this intuitive notion precise. The first mathematical steps were taken in the 19th century by the German Bernhard Riemann and the Italian Enrico Betti, with the introduction of “Betti numbers” in each dimension, referring to the number of independent (suitably defined) objects in that dimension that are not boundaries. Informally, Betti numbers refer to the number of times that an object can be “cut” before splitting into separate pieces; for example, a sphere has Betti number 0 since any cut will split it in two, while a cylinder has Betti number 1 since a cut along its longitudinal axis will merely result in a rectangle. A more extensive treatment of homology was carried out in n dimensions at the beginning of the 20th century by the French mathematician Henri Poincaré, leading to the notion of a homology group in each dimension, apparently first formulated about 1925 by the German mathematician Emmy Noether. The two basic facts about homology groups for a surface or a higher-dimensional topological manifold are: (1) if the groups are defined by means of a triangulation, a cellular subdivision, or other artifact, the resulting groups do not depend on the particular choices made along the way; and (2) the homology groups are a topological invariant, so that if two surfaces or higher-dimensional spaces are homeomorphic, then their homology groups in each dimension are isomorphic (see foundations of mathematics: Isomorphic structures and mathematics: Algebraic topology).

Homology plays a fundamental role in analysis; indeed, Riemann was led to it by questions involving integration on surfaces. The basic reason is because of Green’s theorem (see George Green) and its generalizations, which express certain integrals over a domain in terms of integrals over the boundary. As a consequence, certain important integrals over curves will have the same value for any two curves that are homologous. This is in turn reflected in physics in the study of conservative vector spaces and the existence of potentials.