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.

Given a real vector bundle *E* over *M*, its *k*-th Pontryagin class is defined as

where:

- denotes the -th Chern class of the complexification of
*E*, - is the -cohomology group of
*M*with integer coefficients.

The rational Pontryagin class is defined to be the image of in , the -cohomology group of *M* with rational coefficients.

The **total Pontryagin class**

is (modulo 2-torsion) multiplicative with respect to Whitney sum 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 classes 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 .) 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 *w*_{9} = *w*_{1}*w*_{8} + Sq^{1}(*w*_{8}). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of *E*_{10} with any trivial bundle remains nontrivial. ( Hatcher 2009 , p. 76)

Given a 2*k*-dimensional vector bundle *E* we have

where *e*(*E*) denotes the Euler class of *E*, and denotes the cup product of cohomology classes.

As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes

can be presented as differential forms which depend polynomially on the curvature form of a vector bundle. This Chern–Weil theory revealed a major connection between algebraic topology and global differential geometry.

For a vector bundle *E* over a *n*-dimensional differentiable manifold *M* equipped with a connection, the total Pontryagin class is expressed as

where Ω denotes the curvature form, and *H**_{dR}(*M*) denotes the de Rham cohomology groups.^{[ citation needed ]}

The **Pontryagin classes of a smooth manifold** are defined to be the Pontryagin classes of its tangent bundle.

Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes *p _{k}*(

If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin 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

^{ [1] }

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.

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.^{ [2] }

**Pontryagin numbers** are certain topological invariants of a smooth manifold. 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 *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 of *M*.

- Pontryagin numbers are oriented cobordism 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 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.

There is also a *quaternionic* Pontryagin class, for vector bundles with quaternion structure.

In mathematics, the **Hodge conjecture** is a major unsolved problem in algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 for whoever can prove or disprove the Hodge conjecture.

In mathematics, **K-theory** is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

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 manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc.

In mathematics, and especially differential geometry and gauge theory, a **connection** on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a **linear connection** on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a *covariant derivative*, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, **coherent sheaves** are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In mathematics, **Kähler differentials** provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

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 sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 to *n*, where *n* is the rank of the vector bundle. If the Stiefel–Whitney class of index *i* is nonzero, then there cannot exist (*n*−*i*+1) everywhere linearly independent sections of the vector bundle. A nonzero *n*th Stiefel–Whitney class indicates that every section of the bundle must vanish at some point. A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, *S*^{1}×**R**, is zero.

In mathematics, the **Chern–Weil homomorphism** is a basic construction in **Chern–Weil theory** that computes topological invariants of vector bundles and principal bundles on a smooth manifold *M* in terms of connections and curvature representing classes in the de Rham cohomology rings of *M*. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes.

In mathematics, specifically in algebraic geometry, the **Grothendieck–Riemann–Roch theorem** is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

In mathematics, the **Todd class** is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle.

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 a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, the **adjunction formula** relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction.

In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In mathematics, the **Euler sequence** is a particular exact sequence of sheaves on *n*-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an (*n* + 1)-fold sum of the dual of the Serre twisting sheaf.

In mathematics, the **Segre class** is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by Beniamino Segre (1953). In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton, Segre classes play a fundamental role.

In mathematics, a **projective bundle** is a fiber bundle whose fibers are projective spaces.

In physics and mathematics, and especially differential geometry and gauge theory, the **Yang–Mills equations** are a system of partial differential equations for a connection on a vector bundle or principal bundle. The Yang–Mills equations arise in physics as the Euler–Lagrange equations of the **Yang–Mills action functional**. However, the Yang–Mills equations have independently found significant use within mathematics.

In mathematics, and in particular gauge theory and complex geometry, a **Hermitian Yang–Mills connection** is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called **instantons**.

In algebraic geometry and differential geometry, the **Nonabelian Hodge correspondence** or **Corlette–Simpson correspondence** is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.

In mathematics, and especially differential geometry and mathematical physics, **gauge theory** is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics *theory* means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a physical model of some natural phenomenon.

- ↑ Mclean, Mark. "Pontryagin Classes" (PDF). Archived (PDF) from the original on 2016-11-08.
- ↑ "A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF). p. 16. Archived (PDF) from the original on 2016-01-22.

- Milnor John W.; Stasheff, James D. (1974).
*Characteristic classes*.*Annals of Mathematics Studies*. Princeton, New Jersey; Tokyo: Princeton University Press / University of Tokyo Press. ISBN 0-691-08122-0. - Hatcher, Allen (2009). "Vector Bundles & K-Theory" (2.1 ed.).Cite journal requires
`|journal=`

(help)

- "Pontryagin class",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.