Cohomology

In mathematics, specifically in 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: X → Y 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.

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Definition

For a topological space X, the cohomology group Hn(X;G), with coefficents in G, is defined to be the quotient Ker(δn)/Im(δn-1) at Cn(X;G) in the cochain complex

\cdots \leftarrow C^{n%2B1}(X;G)\ \stackrel{ \delta^n}{\leftarrow}\ C^n (X;G)\ \stackrel{\delta^{n-1}}{\leftarrow}\ C^{n-1}(X;G) \leftarrow \cdots \leftarrow C^0 (X;G) \leftarrow 0.

Elements in Ker(δn) are n-cocycles and elements in Im(δn-1) are n-coboundaries. The cohomology groups with n ≥ 1 are called the higher cohomology.

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.

A generalized cohomology theory is "determined by its values on a point", in the sense that if one has a space given by contractible spaces (homotopy equivalent to a point), glued together along contractible spaces, as in a simplicial complex, then the cohomology of the space is determined by the cohomology of a point and the combinatorics of the patching, and effectively computable. Formally, this is computed by the excision theorem, or equivalently the Mayer–Vietoris sequence. Thus the cohomology of a point is a fundamental calculation for any generalized cohomology theory, though the cohomology of particular spaces is also of interest.

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]

See also

Notes

  1. ^ Spanier, E. H. (2000) "Book reviews: Foundations of Algebraic Topology" Bulletin of the American Mathematical Society 37(1): pp. 114–115
  2. ^ http://www.webcitation.org/query?url=http://www.geocities.com/jefferywinkler2/ktheory3.html&date=2009-10-26+00:45:56

References