homologia

Homologia, or homology, is a term used in several disciplines to denote a relation of similarity arising from common ancestry or from a shared mathematical structure.

In biology, homologous features reflect genetic relatedness. Homologous chromosomes are chromosome pairs in diploid organisms that carry the same genes at the same loci, inherited from each parent; they separate during meiosis. Homologous structures are anatomical features in different species that share a common evolutionary origin, such as the forelimbs of mammals, despite different functions. This is distinguished from analogous structures, which serve similar roles but arise independently through convergent evolution.

In algebraic topology, homology assigns to a space X a sequence of abelian groups H_n(X) that count n‑dimensional holes. They are computed from a chain complex C_n with boundary maps ∂_n: C_n → C_{n-1}, with cycles Z_n = ker ∂_n and boundaries B_n = im ∂_{n+1}, and H_n(X) = Z_n/B_n. Examples: H_0(X) ≅ Z for a connected space; H_1(S^1) ≅ Z; H_2(S^2) ≅ Z. Various theories exist—singular, simplicial, Čech—sharing the same axioms. Homology is functorial and invariant under homotopy, and it has applications in geometry, combinatorics, and data analysis (e.g., persistent homology).

Historically, homology theory developed in the late 19th and early 20th centuries, with foundational contributions by Henri Poincaré and later formalization by Eilenberg and Steenrod.

In chemistry, the term also appears in homologous series, where members differ by repeating units such as CH2.