In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of **cochains**, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the *chains* of homology theory.

From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century; from the initial idea of *homology* as a topologically invariant relation on *chains*, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra. The terminology tends to mask the fact that in many applications *cohomology*, a contravariant theory, is more natural than *homology*. At a basic level this has to do with functions and pullbacks in geometric situations: given spaces *X* and *Y*, and some kind of function *F* on *Y*, for any mapping ƒ: *X* → *Y* composition with ƒ gives rise to a function *F* o ƒ on *X*. Cohomology groups often also have a natural product, the cup product, which gives them a ring structure. Because of this feature, cohomology is a stronger invariant than homology, as it can differentiate between certain algebraic objects that homology cannot.

Read more about Cohomology: Definition, History

### Other articles related to "cohomology":

**Cohomology**Theories - Other

**Cohomology**Theories

... Theories in a broader sense of

**cohomology**include André–Quillen

**cohomology**BRST

**cohomology**Bonar–Claven

**cohomology**Bounded

**cohomology**Coherent

**cohomology**Crystalline

**cohomology**Cyclic

**cohomology**Deligne ...

... Part III, Group

**Cohomology**Abelian Nonabelian

**Cohomology**,

**Cohomology**of Finite Groups, Theorems of Tate and Nakayama, Galois

**Cohomology**, Class Formations, and ...

**Cohomology**With Compact Support

... In mathematics,

**cohomology**with compact support refers to certain

**cohomology**theories, usually with some condition requiring that cocycles should have compact support ...

**Cohomology**

... The group

**cohomology**of a representation M is given by where k has a trivial G-action, so in this way the stable module category gives a natural setting in which group

**cohomology**lives ... Furthermore, the above isomorphism suggests defining

**cohomology**groups for negative values of n, and in this way, one recovers Tate

**cohomology**...

... First, one pulls back the sheaf

**cohomology**groups Hr(Z,F) to the sheaf

**cohomology**Hr(Y,η−1F) on Y in many cases where the Penrose transform is of interest, this ... One then pushes the resulting

**cohomology**classes down to X that is, one investigates the direct image of a

**cohomology**class by means of the Leray spectral sequence ...