In mathematics, specifically in homology theory and algebraic topology, cohomology is a way of attaching algebraic invariants to a topological space or other mathematical object that encode its properties in a way that is often computable. Cohomology is often related to questions of whether some local property of the space is obstructed when passing to a global property. For example, the Möbius strip is not the product space of a line segment with a circle (i.e., a cylinder), but locally on any segment of the circle it resembles an ordinary rectangle (product of two line segments) without the global twist. The obstruction to making the product structure global is encoded in the first cohomology of the underlying circle, which classifies the two inequivalent ways a line can twist around a circle (an even number of twists or an odd number of twists). The fact that there are only two inequivalent ways of twisting a line around a circle is encoded in the relevant cohomology group which can be computed as isomorphic to the group where corresponds to an even number of twists, and to an odd number.
In its general abstract formulation, cohomology is a sequence of abelian groups 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.
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.
Overview
Cohomology is a family of related constructions that occur in several areas of mathematics. The common pattern is that one has objects called cochains, a coboundary operator measuring the failure of a cochain to satisfy a compatibility condition, and cohomology classes obtained by identifying cochains that differ by a trivial or exact contribution.
Example A simple example is the de Rham cohomology of the circle. One considers here smooth functions on the circle, which can be thought of as periodic functions , such that for all . (Here we are parametrizing the functions by radian measure around the unit circle.) The differential of a function is defined by the expression . Differentials of a periodic function have the property that their integral over a whole period is zero: by the fundamental theorem of calculus, , because is periodic over the circle. A differential is sometimes called an exact differential.
Another type of differential can also be considered, which is not necessarily exact. An example is by itself. This does not have an integral of zero over a whole period. On the contrary, . So the differential is inexact. Any differential form , with appropriately periodic is a differential on the circle. It is inexact precisely when , and exact when . Indeed, in the latter case we can write where , the key point being that is periodic (and so defines a function on the circle). Moreover, any differential can be made exact by subtracting from it where .
What we have therefore shown is that, although there are many differentials on the circle (exact or inexact), the quotient space of inexact differentials modulo exact differentials is one-dimensional, parametrized by the constant differential .
Here the coboundaries are the exact differentials and the cochains are all differentials. The quotient of cochains modulo coboundaries is one-dimensional. That is, the first de Rham cohomology group of the circle is isomorphic to the real numbers (as an additive group), .
The circle, being one dimensional, has this one positive cohomology group. Its only other cohomology group consists of functions whose differential is zero, i.e., constants. Cohomology generalizes this basic setup to higher dimensions and other situations.
Cochains and coboundaries Cohomology is often expressed by a cochain complex
with . The -th cohomology group is then
Elements of are called cocycles, elements of are called coboundaries, and elements of the quotient group are called cohomology classes. In the case of the example of the de Rham cohomology of the circle, is all smooth periodic functions, is the differential of a function, is all differentials where is smooth and periodic, and all higher and are zero.
One informal way to understand cohomology is that cocycles are locally or formally consistent data, while coboundaries are data that arise from a more elementary choice. The cohomology group measures the part of the consistent data that cannot be removed in this way. Thus in the case of de Rham cohomology of the circle, a more elementary choice is a function, the formally consistent data is a periodic one-form which is always locally the differential of some function, and the integral of a one-form over the whole circle is the part that cannot be absorbed into the differential of a function. For this reason, cohomology often appears in questions about whether local data can be glued globally, whether a construction can be lifted or extended, or whether two objects that look locally equivalent are globally different. This interpretation is especially explicit in sheaf cohomology, where cohomology measures the failure of local sections of a sheaf to glue to global sections.
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 (the kernel of one homomorphism modulo the image of the previous one). In more detail, is the free abelian group on the set of continuous maps from the standard -simplex to (called "singular -simplices in "), 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 (when coefficients are taken in a field, the dual group is the dual vector space of ), 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 . (Equivalently, an -cochain on can be identified with a function from the set of singular -simplices in to .) Elements of and are called cocycles and coboundaries, respectively, while elements of are called cohomology classes (because they are equivalence classes of cocycles).
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:
A continuous map determines a pushforward homomorphism on homology and a pullback homomorphism on cohomology. This makes cohomology into a contravariant functor from topological spaces to abelian groups (or -modules). Two homotopic maps from to induce the same homomorphism on cohomology (just as on homology). The Mayer–Vietoris sequence is an important computational tool in cohomology, as in homology. Note that the boundary homomorphism increases (rather than decreases) degree in cohomology. That is, if a space is the union of open subsets and , then there is a long exact sequence: There are relative cohomology groups for any subspace of a space . They are related to the usual cohomology groups by a long exact sequence: The universal coefficient theorem describes cohomology in terms of homology, using Ext groups. Namely, there is a short exact sequence A related statement is that for a field , is precisely the dual space of the vector space . If is a topological manifold or a CW complex, then the cohomology groups are zero for greater than the dimension of . If is a compact manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and is a commutative Noetherian ring, then the -module is finitely generated for each .
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 (without boundary), whereas a closed submanifold N of a manifold M means a submanifold that is a closed subset of M, not necessarily compact (although N is automatically compact if M is).
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 [S]. 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 [S][T], 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 (not necessarily compact) 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.