Cohomology theory
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In mathematics, specifically in homology theory and 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. Because of this feature, cohomology is a stronger invariant than homology, as it can differentiate between certain algebraic objects that homology cannot.
Definition
In algebraic topology, the cohomology groups for spaces can be defined as follows (see Hatcher). Given a topological space X, consider the chain complex
as in the definition of singular homology (or simplicial homology). Here, the C_{n} are the free abelian groups generated by formal linear combinations of the singular nsimplices in X and ∂_{n} is the n^{th} boundary operator.
Now replace each C_{n} by its dual space C*_{n−1} = Hom(C_{n}, G), and ∂_{n} by its transpose
to obtain the cochain complex
Then the n^{th} cohomology group with coefficients in G is defined to be Ker(δ^{n+1})/Im(δ^{n}) and denoted by H^{n}(C; G). The elements of C*_{n} are called singular ncochains with coefficients in G , and the δ^{n} are referred to as the coboundary operators. Elements of Ker(δ^{n+1}), Im(δ^{n}) are called cocycles and coboundaries, respectively.
Note that the above definition can be adapted for general chain complexes, and not just the complexes used in singular homology. The study of general cohomology groups was a major motivation for the development of homological algebra, and has since found applications in a wide variety of settings (see below).
Given an element φ of C*_{n1}, it follows from the properties of the transpose that as elements of C*_{n}. We can use this fact to relate the cohomology and homology groups as follows. Every element φ of Ker(δ^{n}) has a kernel containing the image of ∂_{n}. So we can restrict φ to Ker(∂_{n−1}) and take the quotient by the image of ∂_{n} to obtain an element h(φ) in Hom(H_{n}, G). If φ is also contained in the image of δ^{n−1}, then h(φ) is zero. So we can take the quotient by Ker(δ^{n}), and to obtain a homomorphism
It can be shown that this map h is surjective, and that we have a short split exact sequence
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 mid1920s, J. W. Alexander and Solomon Lefschetz founded the intersection theory of cycles on manifolds. On an ndimensional manifold M, a pcycle and a qcycle 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
 H_{p}(M) × H_{q}(M) → H_{p+q−n}(M).
Alexander had by 1930 defined a first cochain notion, based on a pcochain on a space X having relevance to the small neighborhoods of the diagonal in X^{p+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 .
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, , 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:
 simplicial cohomology
 singular cohomology
 de Rham cohomology
 Čech cohomology
Axioms and generalized cohomology theories
There are various ways to define cohomology groups (for example singular cohomology, Čech cohomology, Alexander–Spanier cohomology or Sheaf cohomology). These give different answers for some exotic spaces, but there is a large class of spaces on which they all agree. This is most easily understood axiomatically: there is a list of properties known as the Eilenberg–Steenrod axioms, and any two constructions that share those properties will agree at least on all finite CW complexes, for example.
One of the axioms is the socalled dimension axiom: if P is a single point, then H_{n}(P) = 0 for all n ≠ 0, and H_{0}(P) = Z. We can generalise slightly by allowing an arbitrary abelian group A in dimension zero, but still insisting that the groups in nonzero dimension are trivial. It turns out that there is again an essentially unique system of groups satisfying these axioms, which are denoted by . In the common case where each group H_{k}(X) is isomorphic to Z^{rk} for some r_{k} in N, we just have . In general, the relationship between H_{k}(X) and is only a little more complicated, and is again controlled by the Universal coefficient theorem.
More significantly, we can drop the dimension axiom altogether. There are a number of different ways to define groups satisfying all the other axioms, including the following:
 The stable homotopy groups
 Various different flavours of cobordism groups: and so on. The last of these (known as complex cobordism) is especially important, because of the link with formal group theory via a theorem of Daniel Quillen.
 Various different flavours of Ktheory: (real periodic Ktheory), (real connective), (complex periodic), (complex connective) and so on.
 Brown–Peterson homology, Morava Ktheory, Morava Etheory, and other theories defined using the algebra of formal groups.
 Various flavours of elliptic homology
These are called generalised homology theories; they carry much richer information than ordinary homology, but are often harder to compute. Their study is tightly linked (via the Brown representability theorem) to stable homotopy.
A cohomology theory E is said to be multiplicative if is a graded ring.
Other cohomology theories
Theories in a broader sense of cohomology include:^{}^{}
 André–Quillen cohomology
 BRST cohomology
 Coherent cohomology
 Crystalline cohomology
 Cyclic cohomology
 Deligne cohomology
 Étale cohomology
 Flat cohomology
 Galois cohomology
 Group cohomology
 Hochschild cohomology
 Intersection cohomology
 Khovanov homology
 Lie algebra cohomology
 Local cohomology
 Motivic cohomology
 Nonabelian cohomology
 Quantum cohomology
 Spencer cohomology
See also
 List of cohomology theories
Notes
References
 Hatcher, A. (2001) "Algebraic Topology", Cambridge U press, England: Cambridge, p. 198, ISBN 052179160X and ISBN 0521795400.
 Hazewinkel, M. (ed.), Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia"; Reidel, Dordrecht, Netherlands: 1988; p. 68. ISBN 1556080107
 or see Hazewinkel, Michiel, ed. (2001), "Cohomology", Encyclopedia of Mathematics, Springer, ISBN 9781556080104 .
 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.
