Cohomology

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: XY 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 ...
Local Fields (book) - Contents
... 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 ...
Stable Module Category - Connections With 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 ...
Penrose Transform - Overview
... 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 ...