Cohomology


In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
From its start 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 method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces and, and some function on, for any mapping, composition with gives rise to a function on. The most important cohomology theories have a product, the cup product, which gives them a ring structure. Because of this feature, cohomology is usually a stronger invariant than homology.

Singular cohomology

Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring with any topological space. Every continuous map determines a homomorphism from the cohomology ring of to that of ; this puts strong restrictions on the possible maps from to. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be computable in practice for spaces of interest.
For a topological space, the definition of singular cohomology starts with the singular chain complex:
By definition, the singular homology of is the homology of this chain complex. In more detail, is the free abelian group on the set of continuous maps from the standard -simplex to , and is the -th boundary homomorphism. The groups are zero for negative.
Now fix an abelian group, and replace each group by its dual group and by its dual homomorphism
This has the effect of "reversing all the arrows" of the original complex, leaving a cochain complex
For an integer, the th cohomology group of with coefficients in is defined to be and denoted by. The group is zero for negative. The elements of are called singular -cochains with coefficients in. Elements of and are called cocycles and coboundaries, respectively, while elements of are called cohomology classes.
In what follows, the coefficient group is sometimes not written. It is common to take to be a commutative ring ; then the cohomology groups are -modules. A standard choice is the ring of integers.
Some of the formal properties of cohomology are only minor variants of the properties of homology:
On the other hand, cohomology has a crucial structure that homology does not: for any topological space and commutative ring, there is a bilinear map, called the cup product:
defined by an explicit formula on singular cochains. The product of cohomology classes and is written as or simply as. This product makes the direct sum
into a graded ring, called the cohomology ring of. It is graded-commutative in the sense that:
For any continuous map the pullback is a homomorphism of graded -algebras. It follows that if two spaces are homotopy equivalent, then their cohomology rings are isomorphic.
Here are some of the geometric interpretations of the cup product. In what follows, manifolds are understood to be without boundary, unless stated otherwise. A closed manifold means a compact manifold, whereas a closed submanifold ''N of a manifold M'' means a submanifold that is a closed subset of M, not necessarily compact.
  • Let X be a closed oriented manifold of dimension n. Then Poincaré duality gives an isomorphism HiX ≅ Hn−''iX''. As a result, a closed oriented submanifold S of codimension i in X determines a cohomology class in HiX, called . In these terms, the cup product describes the intersection of submanifolds. Namely, if S and T are submanifolds of codimension i and j that intersect transversally, then where the intersection S ∩ T is a submanifold of codimension i + j, with an orientation determined by the orientations of S, T, and X. In the case of smooth manifolds, if S and T do not intersect transversally, this formula can still be used to compute the cup product , by perturbing S or T to make the intersection transverse. More generally, without assuming that X has an orientation, a closed submanifold of X with an orientation on its normal bundle determines a cohomology class on X. If X is a noncompact manifold, then a closed submanifold determines a cohomology class on X. In both cases, the cup product can again be described in terms of intersections of submanifolds. Note that Thom constructed an integral cohomology class of degree 7 on a smooth 14-manifold that is not the class of any smooth submanifold. On the other hand, he showed that every integral cohomology class of positive degree on a smooth manifold has a positive multiple that is the class of a smooth submanifold. Also, every integral cohomology class on a manifold can be represented by a "pseudomanifold", that is, a simplicial complex that is a manifold outside a closed subset of codimension at least 2.
  • For a smooth manifold X, de Rham's theorem says that the singular cohomology of X with real coefficients is isomorphic to the de Rham cohomology of X, defined using differential forms. The cup product corresponds to the product of differential forms. This interpretation has the advantage that the product on differential forms is graded-commutative, whereas the product on singular cochains is only graded-commutative up to chain homotopy. In fact, it is impossible to modify the definition of singular cochains with coefficients in the integers or in for a prime number p to make the product graded-commutative on the nose. The failure of graded-commutativity at the cochain level leads to the Steenrod operations on mod p cohomology.
Very informally, for any topological space X, elements of can be thought of as represented by codimension-i subspaces of X that can move freely on X. For example, one way to define an element of is to give a continuous map f from X to a manifold M and a closed codimension-i submanifold N of M with an orientation on the normal bundle. Informally, one thinks of the resulting class as lying on the subspace of X; this is justified in that the class restricts to zero in the cohomology of the open subset The cohomology class can move freely on X in the sense that N could be replaced by any continuous deformation of N inside M.

Examples

In what follows, cohomology is taken with coefficients in the integers Z, unless stated otherwise.
  • The cohomology ring of a point is the ring Z in degree 0. By homotopy invariance, this is also the cohomology ring of any contractible space, such as Euclidean space Rn.
  • For a positive integer n, the cohomology ring of the sphere is Z/, with x in degree n. In terms of Poincaré duality as above, x is the class of a point on the sphere.
  • The cohomology ring of the torus is the exterior algebra over Z on n generators in degree 1. For example, let P denote a point in the circle, and Q the point in the 2-dimensional torus. Then the cohomology of 2 has a basis as a free Z-module of the form: the element 1 in degree 0, x := and y := in degree 1, and xy = in degree 2. Note that yx = −xy = −, by graded-commutativity.
  • More generally, let R be a commutative ring, and let X and Y be any topological spaces such that H* is a finitely generated free R-module in each degree. Then the Künneth formula gives that the cohomology ring of the product space X × Y is a tensor product of R-algebras:
  • The cohomology ring of real projective space RPn with Z/2 coefficients is Z/2/, with x in degree 1. Here x is the class of a hyperplane RPn−1 in RPn; this makes sense even though RPj is not orientable for j even and positive, because Poincaré duality with Z/2 coefficients works for arbitrary manifolds. With integer coefficients, the answer is a bit more complicated. The Z-cohomology of RP2a has an element y of degree 2 such that the whole cohomology is the direct sum of a copy of Z spanned by the element 1 in degree 0 together with copies of Z/2 spanned by the elements yi for i=1,...,a. The Z-cohomology of RP2a+1 is the same together with an extra copy of Z in degree 2a+1.
  • The cohomology ring of complex projective space CPn is Z/, with x in degree 2. Here x is the class of a hyperplane CPn−1 in CPn. More generally, xj is the class of a linear subspace CPn−''j in CPn''.
  • The cohomology ring of the closed oriented surface X of genus g ≥ 0 has a basis as a free Z-module of the form: the element 1 in degree 0, A1,...,Ag and B1,...,Bg in degree 1, and the class P of a point in degree 2. The product is given by: AiAj = BiBj = 0 for all i and j, AiBj = 0 if i ≠ j, and AiBi = P for all i. By graded-commutativity, it follows that.
  • On any topological space, graded-commutativity of the cohomology ring implies that 2x2 = 0 for all odd-degree cohomology classes x. It follows that for a ring R containing 1/2, all odd-degree elements of H* have square zero. On the other hand, odd-degree elements need not have square zero if R is Z/2 or Z, as one sees in the example of RP2 or RP4 × RP2.