What does cohomology mean?
Cohomology is a central concept in modern mathematics, particularly algebraic topology, that assigns algebraic objects—usually abelian groups—to topological spaces in a way that captures deep structural information about their shape. Unlike its dual counterpart homology, which measures features such as holes by building them up from chains, cohomology works contravariantly, examining functions defined on those chains and assembling them into cochains. This dual perspective yields extra algebraic structure: cohomology classes can be multiplied together (via the cup product), forming a ring rather than merely a group, which makes it a finer and more flexible invariant. Many important theories fall under this umbrella, including de Rham, Čech, sheaf, and singular cohomology, each suited to different kinds of problems. The word is strictly technical, appearing almost exclusively in advanced mathematical writing, but it plays an indispensable role wherever geometry and algebra intersect.
nounA mathematical tool in algebraic topology that assigns algebraic structures (such as groups or rings) to a topological space, providing invariants that capture essential features of its shape; dually related to homology via the universal coefficient theorem.
- In mathematics: a contravariant functor assigning abelian groups (or modules/rings) to a topological space or more general object, satisfying the Eilenberg–Steenrod axioms, used to distinguish spaces and study their structure.
"The de Rham cohomology of the manifold revealed that its top and bottom Betti numbers agreed."
"Computing the Čech cohomology of the covering showed that H¹ vanished, confirming the bundle was trivial."
"Students first meet singular cohomology as the dual theory to simplicial homology, where cochains are linear functionals on chains."
Cohomology turns the shape of a space into algebra — so powerful that proving the impossibility of certain geometric problems often boils down to computing a single group.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of cohomology
The term combines the prefix 'co-' (indicating duality or a contravariant counterpart) with 'homology', which derives from the Greek 'homos' meaning 'same' and 'logos' meaning 'word' or 'reason'. 'Homology' entered English through earlier biological and mathematical usage before being adopted in topology in the late nineteenth century. The specifically topological sense of 'cohomology' was developed during the 1920s and 1930s by mathematicians working on duality phenomena in algebraic topology.
Related word forms
How cohomology is actually used
Strictly a technical term of algebraic topology and homological algebra; it is countable when referring to specific groups (e.g., 'the second cohomology group') and uncountable when referring to the theory as a whole.
Easily confused with cohomology
Homology assigns chains of simplices measuring 'holes' directly, while cohomology is its dual, assigning cochains (functions on those chains) and yielding a ring structure under cup product.