9.4 Pontryagin classes

Pontryagin classes are key invariants in algebraic topology and differential geometry. They measure the twisting of real vector bundles, providing insights into the structure of manifolds and their associated bundles.

These classes have important properties like naturality and the Whitney sum formula. They're used in cobordism theory, characteristic numbers, and obstruction theory, playing a crucial role in understanding manifold topology and geometric structures.

Definition of Pontryagin classes

• Pontryagin classes are characteristic classes associated to real vector bundles, providing important invariants in algebraic topology and differential geometry
• They are cohomology classes that measure the twisting and non-triviality of a vector bundle, similar to how Chern classes measure the twisting of complex vector bundles
• Pontryagin classes are named after Russian mathematician Lev Pontryagin, who introduced them in the 1940s as part of his work on cobordism theory and the classification of manifolds

Pontryagin classes for real vector bundles

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• For a real vector bundle $E \to B$ of rank $n$, the Pontryagin classes are cohomology classes $p_i(E) \in H^{4i}(B; \mathbb{Z})$ for $i = 1, \ldots, \lfloor \frac{n}{2} \rfloor$
• The first Pontryagin class $p_1(E)$ is the most fundamental and is defined for any real vector bundle of rank at least 2
• Higher Pontryagin classes $p_i(E)$ for $i > 1$ are only defined for vector bundles of rank at least $4i$
• The Pontryagin classes are topological invariants of the vector bundle and do not depend on the choice of connection or metric on the bundle

Pontryagin classes in terms of Chern classes

• For a real vector bundle $E$, its complexification $E \otimes \mathbb{C}$ is a complex vector bundle, and the Pontryagin classes of $E$ can be expressed in terms of the Chern classes of $E \otimes \mathbb{C}$
• The $i$-th Pontryagin class of $E$ is given by $p_i(E) = (-1)^i c_{2i}(E \otimes \mathbb{C})$, where $c_{2i}$ denotes the $2i$-th Chern class
• This relationship allows for the computation of Pontryagin classes using the well-developed theory of Chern classes for complex vector bundles

Formal definition using characteristic classes

• Pontryagin classes are formally defined as characteristic classes associated to the orthogonal group $O(n)$
• They arise from the cohomology of the classifying space $BO(n)$, which classifies real vector bundles of rank $n$ up to isomorphism
• The $i$-th Pontryagin class is the pullback of a universal class $p_i \in H^{4i}(BO(n); \mathbb{Z})$ under the classifying map of a vector bundle
• This definition allows for the study of Pontryagin classes using the tools of homotopy theory and classifying spaces

Properties of Pontryagin classes

• Pontryagin classes satisfy several important properties that make them useful invariants in algebraic topology and differential geometry
• These properties include naturality, the Whitney sum formula, behavior under tensor products and pullbacks, and relations to other characteristic classes
• Understanding these properties is crucial for computing Pontryagin classes in specific examples and applying them to problems in topology and geometry

Naturality of Pontryagin classes

• Pontryagin classes are natural with respect to vector bundle morphisms, meaning they commute with pullbacks
• If $f: E_1 \to E_2$ is a morphism of real vector bundles over a map $g: B_1 \to B_2$, then $f^*(p_i(E_2)) = p_i(E_1)$, where $f^*$ is the induced map on cohomology
• This naturality property allows Pontryagin classes to be used as functorial invariants of vector bundles

Whitney sum formula for Pontryagin classes

• The Pontryagin classes satisfy a Whitney sum formula, which relates the Pontryagin classes of a direct sum of vector bundles to the Pontryagin classes of the individual summands
• For real vector bundles $E$ and $F$ over the same base space, the total Pontryagin class of the direct sum $E \oplus F$ is given by $p(E \oplus F) = p(E) \cdot p(F)$, where $\cdot$ denotes the cup product in cohomology
• This formula allows for the computation of Pontryagin classes of vector bundles built from simpler ones

Pontryagin classes of tensor products

• The Pontryagin classes of a tensor product of real vector bundles can be expressed in terms of the Pontryagin classes of the factors
• For real vector bundles $E$ and $F$ over the same base space, the total Pontryagin class of the tensor product $E \otimes F$ satisfies a certain formula involving the Pontryagin classes of $E$ and $F$
• This formula is more complicated than the Whitney sum formula and involves the splitting principle and symmetric polynomials

Pontryagin classes of pullbacks

• Pontryagin classes behave well under pullbacks of vector bundles
• If $f: B_1 \to B_2$ is a continuous map and $E$ is a real vector bundle over $B_2$, then the Pontryagin classes of the pullback bundle $f^*(E)$ over $B_1$ are given by $p_i(f^*(E)) = f^*(p_i(E))$
• This property allows for the computation of Pontryagin classes of induced bundles and is useful in many applications

Computation of Pontryagin classes

• Computing Pontryagin classes explicitly for specific vector bundles is an important problem in algebraic topology and differential geometry
• There are several methods and techniques for computing Pontryagin classes, including using classifying spaces, characteristic classes, and the relationship with Chern classes
• Some important examples include the Pontryagin classes of projective spaces, Grassmannians, and flag manifolds

Pontryagin classes of projective spaces

• The real projective space $\mathbb{RP}^n$ is the quotient of $\mathbb{R}^{n+1} \setminus \{0\}$ by the equivalence relation identifying points that differ by a scalar multiple
• The tautological line bundle $\gamma^1$ over $\mathbb{RP}^n$ has total Pontryagin class $p(\gamma^1) = 1 + a^2$, where $a \in H^2(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z})$ is the generator of the cohomology ring
• The Pontryagin classes of the tangent bundle of $\mathbb{RP}^n$ can be computed using the Whitney sum formula and the relation between the tangent and normal bundles

Pontryagin classes of Grassmannians

• The Grassmannian $Gr(k, n)$ is the space of $k$-dimensional linear subspaces of $\mathbb{R}^n$
• The tautological vector bundle $\gamma^k$ over $Gr(k, n)$ has Pontryagin classes that generate the cohomology ring of the Grassmannian
• The Pontryagin classes of $\gamma^k$ can be computed using the splitting principle and the relationship between Pontryagin and Chern classes
• The Pontryagin classes of the tangent bundle of $Gr(k, n)$ can be expressed in terms of the Pontryagin classes of $\gamma^k$ and its orthogonal complement

Pontryagin classes of flag manifolds

• A flag manifold is a homogeneous space of the form $G/T$, where $G$ is a compact Lie group and $T$ is a maximal torus in $G$
• Flag manifolds have a rich geometric and topological structure, and their cohomology rings can be described using Schubert calculus
• The Pontryagin classes of the tangent bundles of flag manifolds can be computed using the Borel-Hirzebruch formula, which expresses them in terms of the roots of $G$ and the Weyl group
• Examples of flag manifolds include the complete flag manifold $Fl(n)$ and the partial flag manifolds $Fl(n_1, \ldots, n_k; n)$

Examples of computing Pontryagin classes

• The Pontryagin classes of the tautological line bundle over $\mathbb{RP}^n$ are $p(\gamma^1) = 1 + a^2$, where $a$ is the generator of $H^*(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z})$
• The Pontryagin classes of the tangent bundle of the complex projective space $\mathbb{CP}^n$ are $p(T\mathbb{CP}^n) = (1 + x^2)^{n+1}$, where $x$ is the generator of $H^*(\mathbb{CP}^n; \mathbb{Z})$
• The Pontryagin classes of the tautological vector bundle $\gamma^k$ over the Grassmannian $Gr(k, n)$ can be expressed in terms of the Chern classes of its complexification
• The Pontryagin classes of the tangent bundle of the flag manifold $Fl(n)$ can be computed using the Borel-Hirzebruch formula and the root system of the unitary group $U(n)$

Applications of Pontryagin classes

• Pontryagin classes have numerous applications in algebraic topology, differential geometry, and related fields
• They are used to study the topology of manifolds, the existence of certain geometric structures, and the obstruction to certain constructions
• Some notable applications include cobordism theory, characteristic numbers, obstruction theory, and the study of curvature and characteristic classes in differential geometry

Pontryagin classes and cobordism theory

• Cobordism theory is the study of manifolds up to the equivalence relation of cobordism, where two manifolds are cobordant if their disjoint union is the boundary of a manifold with boundary
• Pontryagin classes play a crucial role in the computation of cobordism groups, which are important invariants in algebraic topology
• The Pontryagin numbers, obtained by evaluating products of Pontryagin classes on the fundamental class of a manifold, are cobordism invariants and provide a powerful tool for distinguishing non-cobordant manifolds

Pontryagin classes and characteristic numbers

• Characteristic numbers are topological invariants of manifolds obtained by integrating certain cohomology classes over the fundamental class of the manifold
• Pontryagin numbers, which are characteristic numbers involving Pontryagin classes, are important invariants in the study of smooth manifolds
• The signature theorem of Hirzebruch relates the signature of a manifold, a important topological invariant, to a certain combination of Pontryagin numbers
• The $\hat{A}$-genus, another important invariant in index theory, can also be expressed in terms of Pontryagin classes

Pontryagin classes and obstruction theory

• Obstruction theory is the study of the obstructions to the existence of certain geometric structures or mappings on manifolds
• Pontryagin classes can be used to define obstructions to the existence of certain structures, such as almost complex structures or spin structures
• The vanishing of certain Pontryagin classes is a necessary condition for the existence of such structures
• In some cases, the Pontryagin classes can be used to completely characterize the obstructions and provide a classification of the possible structures

Pontryagin classes in differential geometry

• In differential geometry, Pontryagin classes are closely related to the curvature of a Riemannian manifold
• The Pontryagin forms, which represent the Pontryagin classes in de Rham cohomology, can be expressed in terms of the curvature tensor of the manifold
• The Gauss-Bonnet theorem and its generalizations relate the Euler characteristic of a manifold to the integral of certain Pontryagin forms
• Pontryagin classes also appear in the study of characteristic classes of foliations and in the Atiyah-Singer index theorem for elliptic operators

Generalizations of Pontryagin classes

• The concept of Pontryagin classes can be generalized and extended in various ways to study more general types of vector bundles and geometric structures
• These generalizations include Pontryagin classes for complex and symplectic vector bundles, Pontryagin classes for foliations, and higher Pontryagin classes and characteristic classes
• These generalizations provide a broader framework for studying the topology and geometry of manifolds and vector bundles

Pontryagin classes for complex vector bundles

• While Pontryagin classes are primarily defined for real vector bundles, they can also be defined for complex vector bundles
• For a complex vector bundle $E$, the Pontryagin classes are defined as the Chern classes of the underlying real vector bundle of $E$
• The Pontryagin classes of a complex vector bundle satisfy similar properties to those of real vector bundles, such as naturality and the Whitney sum formula
• The study of Pontryagin classes for complex vector bundles is closely related to the theory of Chern classes and has applications in complex geometry and topology

Pontryagin classes for symplectic vector bundles

• A symplectic vector bundle is a real vector bundle equipped with a non-degenerate skew-symmetric bilinear form on each fiber
• Pontryagin classes can be defined for symplectic vector bundles and have properties similar to those of Pontryagin classes for real vector bundles
• The Pontryagin classes of a symplectic vector bundle are related to the Chern classes of its complexification and the Euler class of its underlying oriented vector bundle
• The study of Pontryagin classes for symplectic vector bundles has applications in symplectic geometry and topology, such as the classification of symplectic manifolds

Pontryagin classes for foliations

• A foliation on a manifold is a decomposition of the manifold into immersed submanifolds (leaves) of the same dimension
• Pontryagin classes can be defined for the normal bundle of a foliation, which is a vector bundle over the manifold whose fibers are the normal spaces to the leaves
• The Pontryagin classes of the normal bundle of a foliation provide information about the transverse geometry of the foliation and the topology of the ambient manifold
• The study of Pontryagin classes for foliations has applications in the theory of characteristic classes for foliations and the index theory of transversely elliptic operators

Higher Pontryagin classes and characteristic classes

• The classical Pontryagin classes are just the first examples in a more general theory of higher Pontryagin classes and characteristic classes
• Higher Pontryagin classes are cohomology classes associated to higher-dimensional vector bundles and can be defined using the language of classifying spaces and characteristic classes
• Other examples of characteristic classes related to Pontryagin classes include the Euler class, the Stiefel-Whitney classes, and the Chern-Simons classes
• The study of higher Pontryagin classes and characteristic classes is an active area of research in algebraic topology and differential geometry, with connections to physics and other areas of mathematics