HOMOLOGY

in mathematics, correspondence between elements that play similar roles in distinct geometrical figures or mathematical functions. The concept is based on the algebraic structures associated with the topology of geometric regions. A given region can be associated with a qualitatively equivalent region, or complex, composed of triangles put together in such a way so that they meet only at vertices or along their entire edges (see Figure). Points, lines, and triangles are zero-, one-, and two-dimensional objects, and are called 0-cells, 1-cells, and 2-cells, denoted as E0, E1, and E2, respectively, with subscripts added to distinguish between individual cells. The boundary operator, , is simply a way of specifying the end points of a given line, the sides bounding a given triangle, or the triangles bounding a given tetrahedron (see Equations 1 and 2). Any polynomial the terms of which are n-cells (n = 0, 1, 2, . . .) is called an n-chain. The boundary operator can be used on polynomials as well as on single terms, and the resulting boundary cells of each term are added modulo two (i.e., the sum of any two identical cells is zero; see Equation 3). If the resulting boundary chain adds to zero, as when the end-points of all three sides of a triangle are taken, the original chain is called a cycle, with a numerical prefix to indicate its dimension. All cycles, however, are not boundaries; for example, the cycle described in terms of the three sides of the triangular hole (A) is not the boundary of any region (i.e., there is no 2-chain the boundary of which is this cycle). Two 1-cycles Z1 and Z2 are called homologous if their sum Z1 + Z2 is the boundary of some 2-chain. The class of all cycles homologous to each other is called a homology class. The set of all classes of dimension n is called the nth homology group for the geometric region under study. It is the nature of these groups and the way they differentiate between geometric regions that is studied in algebraic topology. The theory of homology groups was extended from Euclidean figures to arbitrary topological spaces by the Austrian Leopold Vietoris (1927) and the Czech Eduard Cech (1932). in biology, similarity of the structure, physiology, or development of different species of organisms based upon their descent from a common evolutionary ancestor. Homology is contrasted with analogy, which is a functional similarity of structure based not upon common evolutionary origins but upon mere similarity of use. Thus the forelimbs of such widely differing mammals as humans, bats, and deer are homologous; the form of construction and the number of bones in these varying limbs are practically identical, and represent adaptive modifications of the forelimb structure of their common early mammalian ancestors. Analogous structures, on the other hand, can be represented by the wings of birds and of insects; the structures are used for flight in both types of organisms, but they have no common ancestral origin at the beginning of their evolutionary development. A 19th-century British biologist, Sir Richard Owen, was the first to define both homology and analogy in precise terms. When two or more organs or structures are basically similar to each other in construction but are modified to perform different functions, they are said to be serially homologous. An example of this is a bat's wing and a whale's flipper. Both originated in the forelimbs of early mammalian ancestors, but they have undergone different evolutionary modification to perform the radically different tasks of flying and swimming, respectively. Sometimes it is unclear whether similarities in structure in different organisms are analogous or homologous. An example of this is the wings of bats and birds. These structures are homologous in that they are in both cases modifications of the forelimb bone structure of early reptiles. But birds' wings differ from those of bats in the number of digits and in having feathers for flight while bats have none. And most importantly, the power of flight arose independently in these two different classes of vertebrates; in birds while they were evolving from early reptiles, and in bats after their mammalian ancestors had already completely differentiated from reptiles. Thus, the wings of bats and birds can be viewed as analogous rather than homologous upon a more rigorous scrutiny of their morphological differences and evolutionary origins.