# Cohomology

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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 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 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

$\cdots \rightarrow C_n \stackrel{ \partial_n}{\rightarrow}\ C_{n-1} \rightarrow \cdots$

as in the definition of singular homology (or simplicial homology). Here, the Cn are the free abelian groups generated by formal linear combinations of the singular n-simplices in X and ∂n is the nth boundary operator.

Now replace each Cn by its dual space C*n−1 = Hom(Cn, G), and ∂n by its transpose

$\delta^n: C_{n-1}^* \rightarrow C_{n}^*$

to obtain the cochain complex

$\cdots \leftarrow C_{n}^* \stackrel{ \delta^n}{\leftarrow}\ C_{n-1}^* \leftarrow \cdots$

Then the nth cohomology group with coefficients in G is defined to be Ker(δn−1)/Im(δn) and denoted by Hn(C; G). The elements of C*n are called singular n-cochains with coefficients in G , and the δn are referred to as the coboundary operators. Elements of Ker(δn−1), Im(δn) 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*n, it follows from the properties of the transpose that $\delta^n(\varphi) = \varphi \circ \partial_n$ as elements of C*n. We can use this fact to relate the cohomology and homology groups as follows. Every element φ of Ker(δn) 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(Hn, G). If φ is also contained in the image of δn−1, then h(φ) is zero. So we can take the quotient by Ker(δn), and to obtain a homomorphism

$h: H^n (C; G) \rightarrow \text{Hom}(H_n(C),G)$

It can be shown that this map h is surjective, and that we have a short split exact sequence

$0 \rightarrow \ker h \rightarrow H^n(C; G) \stackrel{h}{\rightarrow} \text{Hom}(H_n(C),G) \rightarrow 0$

## 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.

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 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.[1]

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:

### Generalized cohomology theories

See also: List of cohomology theories

When one axiom (the dimension axiom) is relaxed, one obtains the idea of generalized cohomology theory or extraordinary cohomology theory; this allows theories based on K-theory and cobordism theory. There are others, coming from stable homotopy theory. In this context, singular homology is referred to as ordinary homology.

The cohomology of a point is called the coefficients of the theory. The coefficients are very important, and are used to compute the cohomology of other spaces using the Atiyah–Hirzebruch spectral sequence. This implies uniqueness, in the sense that if there is a natural transformation between two generalized cohomology theories, which is an isomorphism for a one point space, then it is an isomorphism for all CW complexes. Nevertheless, unlike the case of ordinary cohomology theories, the coefficients alone do not determine the theory in the sense that there might be more than one theory with given coefficients.

One reason that generalized cohomology theories are interesting is that they are representable functors if one works in a larger category than CW complexes; namely, the category of spectra.

### Other cohomology theories

Theories in a broader sense of cohomology include:[2]

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

## Notes

1. Spanier, E. H. (2000) "Book reviews: Foundations of Algebraic Topology" Bulletin of the American Mathematical Society 37(1): pp. 114–115

## References

• Hatcher, A. (2001) "Algebraic Topology", Cambridge U press, England: Cambridge, p. 198, ISBN 0-521-79160-X [Amazon-US | Amazon-UK] and ISBN 0-521-79540-0 [Amazon-US | Amazon-UK]
• 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 [Amazon-US | Amazon-UK]
• 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

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