Pontryagin classes are key invariants in algebraic topology and differential geometry. They measure the twisting of real , providing insights into the structure of and their associated bundles.
These classes have important properties like naturality and the . 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 measure the twisting of complex vector bundles
Pontryagin classes are named after Russian mathematician , 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→B of rank n, the Pontryagin classes are cohomology classes pi(E)∈H4i(B;Z) for i=1,…,⌊2n⌋
The p1(E) is the most fundamental and is defined for any real vector bundle of rank at least 2
Higher Pontryagin classes pi(E) for i>1 are only defined for vector bundles of rank at least 4i
The Pontryagin classes are 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⊗C is a complex vector bundle, and the Pontryagin classes of E can be expressed in terms of the Chern classes of E⊗C
The i-th Pontryagin class of E is given by pi(E)=(−1)ic2i(E⊗C), where c2i 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 pi∈H4i(BO(n);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:E1→E2 is a morphism of real vector bundles over a map g:B1→B2, then f∗(pi(E2))=pi(E1), 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⊕F is given by p(E⊕F)=p(E)⋅p(F), where ⋅ denotes the 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⊗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:B1→B2 is a continuous map and E is a real vector bundle over B2, then the Pontryagin classes of the pullback bundle f∗(E) over B1 are given by pi(f∗(E))=f∗(pi(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 RPn is the quotient of Rn+1∖{0} by the equivalence relation identifying points that differ by a scalar multiple
The tautological line bundle γ1 over RPn has total Pontryagin class p(γ1)=1+a2, where a∈H2(RPn;Z/2Z) is the generator of the cohomology ring
The Pontryagin classes of the tangent bundle of RPn 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 Rn
The tautological vector bundle γk over Gr(k,n) has Pontryagin classes that generate the cohomology ring of the Grassmannian
The Pontryagin classes of γk 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 γk 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(n1,…,nk;n)
Examples of computing Pontryagin classes
The Pontryagin classes of the tautological line bundle over RPn are p(γ1)=1+a2, where a is the generator of H∗(RPn;Z/2Z)
The Pontryagin classes of the tangent bundle of the complex projective space CPn are p(TCPn)=(1+x2)n+1, where x is the generator of H∗(CPn;Z)
The Pontryagin classes of the tautological vector bundle γk 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 theorem of Hirzebruch relates the signature of a manifold, a important topological invariant, to a certain combination of Pontryagin numbers
The A^-genus, another important invariant in , 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 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
Key Terms to Review (17)
Chern classes: Chern classes are topological invariants associated with complex vector bundles that provide crucial information about the geometry and topology of the underlying space. They capture characteristics like curvature and the way bundles twist and turn, connecting deeply with other concepts like cohomology, characteristic classes, and various forms of K-theory.
Chern-Weil Theory: Chern-Weil theory is a mathematical framework that relates differential geometry and topology, providing a way to construct characteristic classes from connections on vector bundles. It offers a systematic method to compute Chern classes and Pontryagin classes by associating curvature forms to these classes, leading to deeper insights into the topology of manifolds and the properties of vector bundles. This theory plays a crucial role in understanding how geometry influences topological invariants.
Cross Product: The cross product is a binary operation on two vectors in three-dimensional space, resulting in a new vector that is orthogonal to both of the original vectors. This operation is fundamental in various mathematical and physical contexts, as it helps in computing areas of parallelograms, determining torque, and analyzing rotations. Understanding the cross product is essential when working with cohomology operations, applying the Cartan formula, and exploring Pontryagin classes.
Cup product: The cup product is an operation in cohomology that combines two cohomology classes to produce a new cohomology class, allowing us to create a ring structure from the cohomology groups of a topological space. This operation plays a key role in understanding the algebraic properties of cohomology, connecting various concepts such as the cohomology ring, cohomology operations, and the Künneth formula.
Development of characteristic classes: The development of characteristic classes refers to a systematic way of associating algebraic invariants to vector bundles, which provide a means to distinguish between different types of bundles based on their topological properties. These classes help in understanding how a vector bundle behaves under continuous transformations and play a crucial role in various areas of mathematics, including topology and geometry. Characteristic classes offer insights into the nature of vector bundles, especially in relation to curvature and intersection theory.
Differential Topology: Differential topology is the field of mathematics that focuses on the properties and structures of differentiable manifolds, which are spaces that locally resemble Euclidean space and allow for calculus to be performed. This area studies the ways in which these manifolds can be transformed and how their shapes affect their topological properties. The insights from differential topology play a crucial role in understanding advanced concepts like Pontryagin classes and cobordism theory.
Euler class: The Euler class is a characteristic class associated with a real vector bundle, providing a way to quantify the topological properties of the bundle. This class is specifically important in the study of oriented manifolds and relates to how the geometry of the manifold interacts with its topology. The Euler class can reveal significant information about the structure of the underlying space, including its curvature and the existence of certain types of sections.
First pontryagin class: The first Pontryagin class is an important topological invariant associated with smooth, oriented, Riemannian manifolds, and is a characteristic class that captures the curvature properties of the tangent bundle. This class is part of a broader framework for understanding the geometry and topology of manifolds, particularly in relation to their differentiable structures and their classifications. The first Pontryagin class can be represented as an element in the second cohomology group, and it plays a significant role in various areas of mathematics, including index theory and gauge theory.
Index theory: Index theory is a fundamental concept in mathematics that connects the geometry of manifolds with the analysis of differential operators, particularly focusing on the relationship between the topology of a manifold and the solutions to differential equations defined on it. This theory provides tools to understand how features like curvature and topology influence the behavior of these differential operators, and it is intimately linked to various characteristic classes and K-theory.
John Milnor: John Milnor is a prominent American mathematician known for his contributions to differential topology, particularly in the development of concepts like exotic spheres and Morse theory. His work has significantly influenced various fields such as topology, geometry, and algebraic topology, connecting foundational ideas to more advanced topics in these areas.
Lev Pontryagin: Lev Pontryagin was a prominent Russian mathematician known for his significant contributions to algebraic topology, particularly in the development of Pontryagin classes. These classes are characteristic classes associated with vector bundles, providing important topological invariants that help classify and understand the structure of manifolds. His work has deep implications in various areas such as differential geometry, topology, and mathematical physics.
Manifolds: Manifolds are topological spaces that locally resemble Euclidean space, allowing for the generalization of concepts like curves and surfaces to higher dimensions. They provide a framework for studying geometrical properties and structures in a flexible way, and are essential for understanding advanced topics in mathematics, such as differential geometry and algebraic topology.
Second Pontryagin Class: The second Pontryagin class is a characteristic class associated with a smooth manifold that measures the curvature of vector bundles over that manifold. It is specifically defined for real vector bundles and provides significant topological information, such as insights into the manifold's structure and the behavior of its tangent bundle. This class is crucial in understanding the relationship between geometry and topology, especially when discussing the invariants of manifolds.
Signature: In the context of mathematics, particularly in algebraic topology, the signature is an invariant associated with a manifold, typically a 4-manifold. It is calculated from the intersection form of the manifold and provides important information about its topology, such as the difference between the numbers of positive and negative eigenvalues of the intersection matrix. Understanding the signature helps in classifying manifolds and relating them to their geometric properties.
Topological invariants: Topological invariants are properties of a topological space that remain unchanged under continuous deformations, such as stretching or bending, but not tearing or gluing. These invariants help classify spaces and reveal essential features about their structure, playing a crucial role in various mathematical theories and applications.
Vector Bundles: A vector bundle is a mathematical structure that consists of a topological space called the base space, along with a vector space attached to each point of that base space. This concept is vital in understanding how vector spaces can vary smoothly over a manifold, allowing for the examination of geometrical and topological properties. The notion of vector bundles is intricately connected to various theories that assign characteristic classes, providing tools to study the geometric nature of the bundles and their implications on other mathematical structures.
Whitney Sum Formula: The Whitney Sum Formula relates the characteristic classes of vector bundles, providing a way to compute the total Chern class of the sum of two vector bundles in terms of their individual Chern classes. This formula captures how the topology of a space changes when combining vector bundles and allows for connections between different types of characteristic classes, including Wu classes and Pontryagin classes.