In
mathematics, specifically in
homology theory and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, 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 po ...
, often defined from a
cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
. 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
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s 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. A c ...
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 ...
and
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
. 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: i ...
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 In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
, which gives them a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
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 In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy
:xy = (-1)^ yx,
where , ''x'' , and , ''y'' , d ...
with any topological space. Every
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
''f'': ''X'' → ''Y'' determines a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
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
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
:
By definition, the
singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
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
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
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
* Mo ...
s. A standard choice is the ring Z of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s.
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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
from topological spaces to abelian groups (or ''R''-modules).
* Two
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
maps from ''X'' to ''Y'' induce the same homomorphism on cohomology (just as on homology).
* The
Mayer–Vietoris sequence
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due ...
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
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
:
* 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
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'':
:
completely ...
describes cohomology in terms of homology, using
Ext groups. Namely, there is a
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
A related statement is that for a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''F'',
is precisely the
dual space of the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
.
* If ''X'' is a topological
manifold or a
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
, 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 coor ...
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 British ...
manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and ''R'' is a commutative
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
, 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
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
, called the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
:
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
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
, 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: i ...
is a homomorphism of graded ''R''-
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
s. It follows that if two spaces are
homotopy equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
, 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
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
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 manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s, 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
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannian m ...
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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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 homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain c ...
. 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 operation In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.
For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, ...
s 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
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
space, such as Euclidean space R
''n''.
*
![torus cycles](https://upload.wikimedia.org/wikipedia/commons/8/81/Torus_cycles.svg)
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 th ...
is Z
'x''(''x''
2) (the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
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 considere ...
), 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 not tou ...
is the
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
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 Künneth is a surname. Notable people with the surname include:
* Hermann Künneth (1892–1975), German mathematician
* Walter Künneth (1901–1997), German Protestant theologian
{{DEFAULTSORT:Kunneth
German-language surnames ...
gives that the cohomology ring of the
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
''X'' × ''Y'' is a
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of ''R''-algebras:
* The cohomology ring of
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
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
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
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 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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
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
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
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
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
with values in ''H''
''n''(''X'',Z) ≅ Z is a perfect pairing over Z.
Characteristic classes
An oriented real
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
''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 o ...
χ(''E'') ∈ ''H''
''r''(''X'',Z). Informally, the Euler class is the class of the zero set of a general
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
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 class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es for vector bundles that take values in cohomology, including
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
es,
Stiefel–Whitney class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
es, 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 bundl ...
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