Homology

Homology is a central concept in two distinct fields: mathematics and biology, where it denotes different kinds of correspondence or structure-preserving relations.

In mathematics, homology is an algebraic tool used in topology to study the shape of spaces. It assigns to a topological space X a sequence of abelian groups or modules H_n(X) called the homology groups, defined from a chain complex built from n-simplices or singular simplices. Intuitively, these groups detect holes of various dimensions: H_0 corresponds to connected components, H_1 to loops that cannot be contracted to a point, H_2 to voids enclosed by surfaces, and higher H_n to higher-dimensional voids. There are several concrete versions, such as singular, simplicial, and cellular homology, which lead to the same invariants for appropriate spaces. Homology is functorial, is invariant under homeomorphisms (indeed under homotopy equivalence), and satisfies exact sequences that facilitate computations, notably the Mayer–Vietoris sequence and the Kunneth formula for products. Reduced homology adjusts H_0 to better reflect connectedness, and various computational tools enable calculation for typical spaces.

In biology, homology refers to similarity due to common ancestry. Homologous structures share a developmental origin and may perform different functions, as in the forelimbs of mammals, birds, and reptiles. This contrasts with analogy, which arises from convergent evolution and independent origins. In genetics, homologous chromosomes carry corresponding genes; homologs can be orthologs (across species) or paralogs (within a genome). The concept helps trace evolutionary relationships and understand conserved features across organisms.