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 ...
, the Pontryagin classes, named after
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
, are certain
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 of real vector bundles. The Pontryagin classes lie in
cohomology group
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 viewe ...
s with degrees a multiple of four.
Definition
Given a real vector bundle ''E'' over ''M'', its ''k''-th Pontryagin class
is defined as
:
where:
*
denotes the
-th
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 ma ...
of the
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...
of ''E'',
*
is the
-
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 viewe ...
group of ''M'' with
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 language ...
coefficients.
The rational Pontryagin class
is defined to be the image of
in
, the
-cohomology group of ''M'' with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
coefficients.
Properties
The total Pontryagin class
:
is (modulo 2-torsion) multiplicative with respect to
Whitney sum
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 po ...
of vector bundles, i.e.,
:
for two vector bundles ''E'' and ''F'' over ''M''. In terms of the individual Pontryagin classes ''p
k'',
:
:
and so on.
The vanishing of the Pontryagin classes and
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 of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to
vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle
over the
9-sphere. (The
clutching function for
arises from the
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class ''w''
9 of ''E''
10 vanishes by the
Wu formula
Wu may refer to:
States and regions on modern China's territory
*Wu (state) (; och, *, italic=yes, links=no), a kingdom during the Spring and Autumn Period 771–476 BCE
** Suzhou or Wu (), its eponymous capital
** Wu County (), a former county i ...
''w''
9 = ''w''
1''w''
8 + Sq
1(''w''
8). Moreover, this vector bundle is stably nontrivial, i.e. the
Whitney sum
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 po ...
of ''E''
10 with any trivial bundle remains nontrivial.
Given a 2''k''-dimensional vector bundle ''E'' we have
:
where ''e''(''E'') denotes 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 of ...
of ''E'', and
denotes 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 commutat ...
of cohomology classes.
Pontryagin classes and curvature
As was shown by
Shiing-Shen Chern
Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...
and
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
around 1948, the rational Pontryagin classes
:
can be presented as differential forms which depend polynomially on the
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algebra ...
of a vector bundle. This
Chern–Weil theory revealed a major connection between algebraic topology and global differential geometry.
For a
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 po ...
''E'' over a ''n''-dimensional
differentiable 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 ...
''M'' equipped with a
connection, the total Pontryagin class is expressed as
:
where Ω denotes the
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algebra ...
, and ''H*''
dR(''M'') denotes the
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
groups.
Pontryagin classes of a manifold
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
.
Novikov Novikov, Novikoff (masculine, russian: Новиков) or Novikova (feminine, russian: Новикова) is one of the most common Russian surnames. Derived from '' novik'' - a teenager on military service who comes from a noble, boyar or cossack ...
proved in 1966 that if two compact, oriented, smooth manifolds are
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
then their rational Pontryagin classes ''p
k''(''M'', Q) in ''H''
4''k''(''M'', Q) are the same.
If the dimension is at least five, there are at most finitely many different smooth manifolds with given
homotopy type
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 ...
and Pontryagin classes.
Pontryagin classes from Chern classes
The Pontryagin classes of a complex vector bundle
can be completely determined by its Chern classes. This follows from the fact that
, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is,
and
. Then, this given the relation
for example, we can apply this formula to find the Pontryagin classes of a vector bundle on a curve and a surface. For a curve, we have
so all of the Pontryagin classes of complex vector bundles are trivial. On a surface, we have
showing
. On line bundles this simplifies further since
by dimension reasons.
Pontryagin classes on a Quartic K3 Surface
Recall that a quartic polynomial whose vanishing locus in
is a smooth subvariety is a K3 surface. If we use the normal sequence
we can find
showing
and
. Since
corresponds to four points, due to Bezout's lemma, we have the second chern number as
. Since
in this case, we have
. This number can be used to compute the third stable homotopy group of spheres.
Pontryagin numbers
Pontryagin numbers are certain
topological invariant
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces ...
s of a smooth
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. Each Pontryagin number of a manifold ''M'' vanishes if the dimension of ''M'' is not divisible by 4. It is defined in terms of the Pontryagin classes of the
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''M'' as follows:
Given a smooth
-dimensional manifold ''M'' and a collection of natural numbers
:
such that
,
the Pontryagin number
is defined by
:
where
denotes the ''k''-th Pontryagin class and
'M''the
fundamental class
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundamen ...
of ''M''.
Properties
#Pontryagin numbers are oriented
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...
invariant; and together with
Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
#Pontryagin numbers of closed Riemannian manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
#Invariants such as
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
and
-genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see
Hirzebruch signature theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem)
is Friedrich Hirzebruch's 1954 result expressing the signature
of a smooth closed oriented manifold by a linear combi ...
.
Generalizations
There is also a ''quaternionic'' Pontryagin class, for vector bundles with
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
structure.
See also
*
Chern–Simons form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from whic ...
*
Hirzebruch signature theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem)
is Friedrich Hirzebruch's 1954 result expressing the signature
of a smooth closed oriented manifold by a linear combi ...
References
*
*
External links
* {{springer, title=Pontryagin class, id=p/p073750
Characteristic classes
Differential topology