# User Contributed Dictionary

- A theory associating a system of quotient groups to each topological space.
- A system of quotient groups associated to a topological space.

# Extensive Definition

In mathematics, specifically in
algebraic
topology, cohomology is a general term for a sequence of abelian
groups defined from a cochain
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 f : X → Y composition
with f gives rise to a function F o f on X. Cohomology groups often
also have a natural product, the cup product,
which gives them a ring
structure.

With hindsight, general homology
theory should probably have been given an inclusive meaning
covering both homology and cohomology: the direction of the arrows
in a chain
complex is not much more than a sign
convention.

## History

Although cohomology is fundamental to modern
algebraic
topology, its importance was not seen for some 40 years after
the development of homology. The concept of dual cell structure,
which Henri
Poincaré used in his proof of his Poincaré
duality theorem, contained the germ of the idea of cohomology,
but this was not seen until later.

There were various precursors to cohomology. In
the mid-1920s, J.W.
Alexander and Solomon
Lefschetz founded the intersection
theory of cycles on manifolds. On an n-dimensional manifold M, a
p-cycle and a q-cycle with nonempty intersection will, if in
general
position, have intersection a (p+q−n)-cycle. This
enables us to define a multiplication of homology classes

- Hp(M) × Hq(M) → Hp+q-n(M).

Alexander had by 1930 defined a first cochain
notion, based on a p-cochain on a space X having relevance to the
small neighborhoods of the diagonal in Xp+1.

In 1931, Georges de
Rham related homology and exterior differential
forms, proving De Rham's
theorem. This result is now understood to be more naturally
interpreted in terms of cohomology.

In 1934, Lev
Pontryagin proved the Pontryagin
duality theorem; a result on topological
groups. This (in rather special cases) provided an
interpretation of Poincaré
duality and Alexander
duality in terms of group
characters.

At a 1935 conference in
Moscow,
Andrey
Kolmogorov and Alexander both introduced cohomology and tried
to construct a cohomology product structure.

From 1936 to 1938, Hassler
Whitney and Eduard
Čech developed the cup product
(making cohomology into a graded ring) and cap product,
and realized that Poincaré duality can be stated in terms of the
cap product. Their theory was still limited to finite
cell complexes.

In 1944, Samuel
Eilenberg overcame the technical limitations, and gave the
modern definition of singular
homology and cohomology.

In 1945, Eilenberg and
Steenrod stated the axioms
defining a homology or cohomology theory. In their 1952 book,
Foundations of Algebraic Topology, they proved that the
existing homology and cohomology theories did indeed satisfy their
axioms.

In 1948 Edwin
Spanier, building on work of Alexander and Kolmogorov,
developed Alexander-Spanier
cohomology.

## Cohomology theories

### Eilenberg-Steenrod theories

A cohomology theory is a family of contravariant
functors from the
category
of pairs of topological
spaces and continuous
functions (or some subcategory thereof such as
the category of CW complexes)
to the category of Abelian
groups and group homomorphisms that
satisfies the Eilenberg-Steenrod
axioms.

Some cohomology theories in this sense are:

### Extraordinary cohomology theories

When one axiom (dimension axiom) is relaxed, one
obtains the idea of extraordinary cohomology theory; this allows
theories based on K-theory and
cobordism
theory. There are others, coming from stable
homotopy theory.

### Other cohomology theories

Theories in a broader sense of cohomology
include:

- Group cohomology
- Galois cohomology
- Lie algebra cohomology
- Harrison cohomology
- Γ cohomology
- Schur cohomology
- André-Quillen cohomology
- Hochschild cohomology
- Cyclic cohomology
- Topological André-Quillen cohomology
- Topological Hochschild cohomology
- Topological Cyclic cohomology
- Coherent cohomology
- Local cohomology
- Étale cohomology
- Crystalline cohomology
- Flat cohomology
- Motivic cohomology
- Deligne cohomology
- Perverse cohomology
- Intersection cohomology
- Non-abelian cohomology
- Gel'fand-Fuks cohomology
- Spencer cohomology
- Bonar-Claven cohomology
- Quantum cohomology

## References

- Hazewinkel, M. (ed.) (1988) Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia" Dordrecht, Netherlands: Reidel, Dordrecht, Netherlands, p. 68, ISBN 1-55608-010-7
- E. Cline, B. Parshall, L. Scott and W. van der Kallen, (1977) "Rational and generic cohomology" Inventiones Mathematicae 39(2): pp. 143–163
- Asadollahi, Javad and Salarian, Shokrollah (2007) "Cohomology theories for complexes" Journal of Pure & Applied Algebra 210(3): pp. 771-787

## See also

cohomology in Spanish: Cohomología

cohomology in French: Homologie et
cohomologie

cohomology in Dutch: Cohomologie

cohomology in Portuguese: Cohomologia

cohomology in Russian: Гомология
(топология)#.D0.9A.D0.BE.D0.B3.D0.BE.D0.BC.D0.BE.D0.BB.D0.BE.D0.B3.D0.B8.D0.B8

cohomology in Finnish:
Kohomologia