In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
homology theory and
algebraic topology, cohomology is a general term for a sequence of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, usually one associated with a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, 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
A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
in homology theory.
From its beginning in
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
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
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
s in geometric situations: given spaces ''X'' and ''Y'', and some kind of function ''F'' on ''Y'', for any
mapping , composition with ''f'' gives rise to a function on ''X''. 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 ''f'': ''X'' → ''Y'' determines a
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from the cohomology ring of ''Y'' to that of ''X''; this puts strong restrictions on the possible maps from ''X'' to ''Y''. 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 ''X'', the definition of singular cohomology starts with the
singular chain complex:
By definition, the
singular homology of ''X'' is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail, ''C
i'' is the
free abelian group on the set of continuous maps from the standard ''i''-simplex to ''X'' (called "singular ''i''-simplices in ''X''"), and ∂
''i'' is the ''i''-th boundary homomorphism. The groups ''C''
''i'' are zero for ''i'' negative.
Now fix an abelian group ''A'', and replace each group ''C
i'' 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 ''i'', the ''i''
th cohomology group of ''X'' with coefficients in ''A'' is defined to be ker(''d
i'')/im(''d''
''i''−1) and denoted by ''H''
''i''(''X'', ''A''). The group ''H''
''i''(''X'', ''A'') is zero for ''i'' negative. The elements of
are called singular ''i''-cochains with coefficients in ''A''. (Equivalently, an ''i''-cochain on ''X'' can be identified with a function from the set of singular ''i''-simplices in ''X'' to ''A''.) Elements of ker(''d'') and im(''d'') are called cocycles and coboundaries, respectively, while elements of ker(''d'')/im(''d'') = ''H''
''i''(''X'', ''A'') are called cohomology classes (because they are
equivalence classes of cocycles).
In what follows, the coefficient group ''A'' is sometimes not written. It is common to take ''A'' to be a
commutative ring ''R''; then the cohomology groups are ''R''-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s. A standard choice is the ring Z 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 ''R''-modules).
* Two
homotopic maps from ''X'' to ''Y'' 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 ''X'' is the union of
open subsets ''U'' and ''V'', then there is a
long exact sequence:
* There are
relative cohomology groups
for any
subspace ''Y'' of a space ''X''. 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 group
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic s ...
s. Namely, there is a
short exact sequence A related statement is that for a
field ''F'',
is precisely the
dual space of the
vector space .
* If ''X'' is a topological
manifold or a
CW complex, then the cohomology groups
are zero for ''i'' greater than the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of ''X''. If ''X'' is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and ''R'' is a commutative
Noetherian ring, then the ''R''-module ''H''
''i''(''X'',''R'') is
finitely generated for each ''i''.
On the other hand, cohomology has a crucial structure that homology does not: for any topological space ''X'' and commutative ring ''R'', there is a
bilinear map, called the
cup product:
defined by an explicit formula on singular cochains. The product of cohomology classes ''u'' and ''v'' is written as ''u'' ∪ ''v'' or simply as ''uv''. This product makes the
direct sum
into a
graded ring, called the
cohomology ring of ''X''. It is
graded-commutative in the sense that:
For any continuous map
the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
is a homomorphism of graded ''R''-
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 ''H''
''i''''X'' ≅ ''H''
''n''−''i''''X''. As a result, a closed oriented submanifold ''S'' of
codimension ''i'' in ''X'' determines a cohomology class in ''H''
''i''''X'', 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
transversely, 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 transversely, 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.
* 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 R
''n''.
*
For a positive integer ''n'', the cohomology ring of the
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
is Z
'x''(''x''
2) (the
quotient ring of a
polynomial ring by the given
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
), 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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
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 (''P'',''P'') in the 2-dimensional torus
. Then the cohomology of (''S''
1)
2 has a basis as a
free Z-module of the form: the element 1 in degree 0, ''x'' :=
1">'P'' × ''S''1and ''y'' :=
1 × ''P''">'S''1 × ''P''in degree 1, and ''xy'' =
'Q''in degree 2. (Implicitly, orientations of the torus and of the two circles have been fixed here.) Note that ''yx'' = −''xy'' = −
'Q'' by graded-commutativity.
*More generally, let ''R'' be a commutative ring, and let ''X'' and ''Y'' be any topological spaces such that ''H''
*(''X'',''R'') is a finitely generated free ''R''-module in each degree. (No assumption is needed on ''Y''.) 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 RP
''n'' with Z/2 coefficients is Z/2
'x''(''x''
''n''+1), with ''x'' in degree 1. Here ''x'' is the class of a
hyperplane RP
''n''−1 in RP
''n''; this makes sense even though RP
''j'' 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 RP
2''a'' 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 ''y''
''i'' for ''i''=1,...,''a''. The Z-cohomology of RP
2''a''+1 is the same together with an extra copy of Z in degree 2''a''+1.
*The cohomology ring of
complex projective space CP
''n'' is Z
'x''(''x''
''n''+1), with ''x'' in degree 2. Here ''x'' is the class of a hyperplane CP
''n''−1 in CP
''n''. More generally, ''x''
''j'' is the class of a linear subspace CP
''n''−''j'' in CP
''n''.
*The cohomology ring of the closed oriented surface ''X'' of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''g'' ≥ 0 has a basis as a free Z-module of the form: the element 1 in degree 0, ''A''
1,...,''A''
''g'' and ''B''
1,...,''B''
''g'' in degree 1, and the class ''P'' of a point in degree 2. The product is given by: ''A''
''i''''A''
''j'' = ''B''
''i''''B''
''j'' = 0 for all ''i'' and ''j'', ''A''
''i''''B''
''j'' = 0 if ''i'' ≠ ''j'', and ''A''
''i''''B''
''i'' = ''P'' for all ''i''. By graded-commutativity, it follows that .
*On any topological space, graded-commutativity of the cohomology ring implies that 2''x''
2 = 0 for all odd-degree cohomology classes ''x''. It follows that for a ring ''R'' containing 1/2, all odd-degree elements of ''H''
*(''X'',''R'') 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 RP
2 (with Z/2 coefficients) or RP
4 × RP
2 (with Z coefficients).
The diagonal
The cup product on cohomology can be viewed as coming from the
diagonal map
In category theory, a branch of mathematics, for any object a in any category \mathcal where the product a\times a exists, there exists the diagonal morphism
:\delta_a : a \rightarrow a \times a
satisfying
:\pi_k \circ \delta_a = \operatorna ...
Δ: ''X'' → ''X'' × ''X'', ''x'' ↦ (''x'',''x''). Namely, for any spaces ''X'' and ''Y'' with cohomology classes ''u'' ∈ ''H''
''i''(''X'',''R'') and ''v'' ∈ ''H''
''j''(''Y'',''R''), there is an external product (or cross product) cohomology class ''u'' × ''v'' ∈ ''H''
''i''+''j''(''X'' × ''Y'',''R''). The cup product of classes ''u'' ∈ ''H''
''i''(''X'',''R'') and ''v'' ∈ ''H''
''j''(''X'',''R'') can be defined as the pullback of the external product by the diagonal:
Alternatively, the external product can be defined in terms of the cup product. For spaces ''X'' and ''Y'', write ''f'': ''X'' × ''Y'' → ''X'' and ''g'': ''X'' × ''Y'' → ''Y'' for the two projections. Then the external product of classes ''u'' ∈ ''H''
''i''(''X'',''R'') and ''v'' ∈ ''H''
''j''(''Y'',''R'') is:
Poincaré duality
Another interpretation of Poincaré duality is that the cohomology ring of a closed oriented manifold is self-dual in a strong sense. Namely, let ''X'' be a closed
connected oriented manifold of dimension ''n'', and let ''F'' be a field. Then ''H''
''n''(''X'',''F'') is isomorphic to ''F'', and the product
:
is a
perfect pairing for each integer ''i''. In particular, the vector spaces ''H''
''i''(''X'',''F'') and ''H''
''n''−''i''(''X'',''F'') have the same (finite) dimension. Likewise, the product on integral cohomology modulo
torsion with values in ''H''
''n''(''X'',Z) ≅ Z is a perfect pairing over Z.
Characteristic classes
An oriented real
vector bundle ''E'' of rank ''r'' over a topological space ''X'' determines a cohomology class on ''X'', the
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle ...
χ(''E'') ∈ ''H''
''r''(''X'',Z). Informally, the Euler class is the class of the zero set of a general
section of ''E''. That interpretation can be made more explicit when ''E'' is a smooth vector bundle over a smooth manifold ''X'', since then a general smooth section of ''X'' vanishes on a codimension-''r'' submanifold of ''X''.
There are several other types of
characteristic classes for vector bundles that take values in cohomology, including
Chern classes,
Stiefel–Whitney classes, and
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Definition
Given a real vector bundle ...
es.
Eilenberg–MacLane spaces
For each abelian group ''A'' and natural number ''j'', there is a space
whose ''j''-th homotopy group is isomorphic to ''A'' and whose other homotopy groups are zero. Such a space is called an Eilenberg–MacLane space. This space has the remarkable property that it is a classifying space for cohomology: there is a natural element ''u'' of
, and every cohomology class of degree ''j'' on every space ''X'' is the pullback of ''u'' by some continuous map
. More precisely, pulling back the class ''u'' gives a bijection
:
for every space ''X'' with the homotopy type of a CW complex. Here