Everything about Cohomology totally explained
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 isn't much more than a
sign convention.
History
Although cohomology is fundamental to modern
algebraic topology, its importance wasn't 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 wasn't 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.
In
1936 Norman Steenrod published a paper constructing
Čech cohomology by dualizing
Čech homology.
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 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 cohomologyFurther Information
Get more info on 'Cohomology'.
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