Pontryagin ClassEdit
Pontryagin classes are a family of topological invariants associated with real vector bundles. Named after Isaak (Iosif) Pontryagin, they live in even-degree cohomology and provide a robust set of obstructions and classifiers for how bundles and the underlying manifolds can twist and turn. For an oriented real vector bundle E over a space M, the Pontryagin classes p_i(E) live in H^{4i}(M; Z) for i ≥ 1, and together they form the total Pontryagin class p(E) = 1 + p_1(E) + p_2(E) + ...
Pontryagin classes are natural under pullback and independent of the choice of connection, making them intrinsic invariants of the bundle. They can be defined in several equivalent ways, including via the complexification E ⊗ C of a real bundle and the associated Chern classes, or through Chern-Weil theory using curvature forms on a chosen connection.
In many places one encounters the compact relation to complex geometry: for a real oriented bundle E, if E_C denotes the complexification, the Pontryagin classes can be expressed in terms of the Chern classes of E_C by p_i(E) = (−1)^i c_{2i}(E_C). Equivalently, the total Pontryagin class can be written in terms of the Chern roots of E_C as p(E) = ∏_j (1 + x_j^2), where the x_j are the Chern roots. This bridges the real and complex viewpoints and explains why Pontryagin classes are detected by curvature in the Chern-Weil framework.
The first Pontryagin class p_1(E) is the most frequently used invariant. In particular, for a closed oriented 4-manifold M, the Hirzebruch signature theorem identifies the manifold’s signature with a Pontryagin number: the signature equals (1/3) ⟨p_1(TM), [M]⟩, where TM is the tangent bundle of M and [M] is the fundamental class. This makes p_1(TM) a critical bridge between geometry and topology on four-dimensional spaces. More generally, Pontryagin numbers are the integers obtained by evaluating polynomials in the p_i on the fundamental class of a smooth closed manifold, and they play a central role in cobordism theory and the classification of smooth structures.
The Pontryagin classes also interact with other characteristic classes. For example, in complex geometry one often compares p_i(E) with Chern classes of the complexified bundle E_C. When E comes from the tangent bundle of a complex manifold, relations among p_i(TM) and the Chern numbers of the complex structure yield constraints and identities that illuminate how complex and real geometry intertwine. In addition, the mod 2 reduction of p_1(TM) obeys the relation p_1(TM) ≡ w_2(TM)^2 (mod 2), tying Pontryagin theory to the Stiefel-Whitney framework for obstruction theory. For spin manifolds, where w_2(TM) = 0, this implies p_1(TM) is even.
The universal aspects of Pontryagin classes are captured by the classifying space BSO(n) for oriented real rank-n bundles. The universal Pontryagin classes live in the cohomology of BSO(n) and pull back to p_i(E) for any E over M via the classifying map M → BSO(n). This viewpoint situates Pontryagin classes within the broader theory of characteristic classes and the geometry of classifying spaces classifying space and BSO(n).
From a differential-geometric standpoint, Pontryagin classes admit concrete representatives as differential forms through Chern-Weil theory. Given a connection on E with curvature F, the Pontryagin forms are produced by invariant polynomials in F, yielding closed 4i-forms that represent p_i(E) in de Rham cohomology. A typical example is p_1 represented (up to a universal constant) by the 4-form Tr(F ∧ F). These forms are independent of the chosen connection, reflecting the topological nature of the classes.
The theory of Pontryagin classes interacts with several major topics in topology and geometry. The L-classes, built from the Pontryagin classes, encode the Hirzebruch signature genus and control the behavior of the signature under various geometric operations. The study of p_i leads to constraints on which manifolds can admit certain types of bundles, and to computations of characteristic numbers that distinguish smooth structures in high dimensions. They also appear in index theory through the A-hat genus and related invariants, tying together analysis, topology, and geometry in a coherent framework.
In summary, the Pontryagin classes form a cornerstone of modern differential topology. They provide a robust algebraic grasp on the twisting of real vector bundles, link real and complex viewpoints via the complexification, encode global geometric information through curvature, and have concrete implications for the topology of manifolds through signatures and Pontryagin numbers.
Definition and basic properties
- Real and oriented bundles: For an oriented real vector bundle E → M of rank n, one can attach a sequence of integral cohomology classes p_i(E) ∈ H^{4i}(M; Z) for i ≥ 1, assembled as the total Pontryagin class p(E) = 1 + p_1(E) + p_2(E) + ... .
- Naturalness: If f: N → M is a continuous map and f^E is the pullback bundle, then p_i(f^*E) = f^ p_i(E).
- Independence from connection: The cohomology classes p_i(E) do not depend on any particular connection on E; they are intrinsic to the isomorphism class of the bundle.
- Relation to complexification: If E_C = E ⊗ C is the complexification of E, then p_i(E) = (−1)^i c_{2i}(E_C). Equivalently, p(E) = ∏_j (1 + x_j^2), where the x_j are the Chern roots of E_C. See also Chern class and Chern-Weil theory.
- First nontrivial class: The class p_1(E) ∈ H^4(M; Z) is the most commonly used invariant, with higher p_i capturing increasingly refined twisting information.
Construction via Chern-Weil theory and universal classes
- Chern-Weil viewpoint: On a chosen connection ∇ on E with curvature F, the Pontryagin forms are constructed from invariant polynomials in F. The resulting forms represent p_i(E) in de Rham cohomology, providing a bridge between differential geometry and topology.
- Universal classes: The classifying space for oriented real bundles, BSO(n), carries universal Pontryagin classes p_i, which pull back to p_i(E) on any base through the classifying map M → BSO(n). This universal perspective situates p_i within the broader framework of characteristic classes and classifying spaces classifying space BSO(n).
Relationships and examples
- Four-manifolds and the signature: For a closed oriented 4-manifold M, the Hirzebruch signature theorem asserts that the signature σ(M) equals (1/3) ⟨p_1(TM), [M]⟩. This identifies p_1(TM) with a fundamental geometric quantity in four dimensions.
- Complex tangents and Chern numbers: If M carries a complex structure and TM is viewed as a complex vector bundle, the relation p_i(TM) = (−1)^i c_{2i}(TM_C) ties the real Pontryagin data to the complex Chern data of the tangent bundle.
- Spin geometry: Since p_1 is related to w_2 through mod 2 reduction (p_1(TM) ≡ w_2(TM)^2 (mod 2)), spin manifolds (where w_2(TM) = 0) impose parity constraints on p_1(TM). This links Pontryagin theory to spin structures and index-theoretic invariants such as the Â-genus.
- Tangent bundles and universal obstructions: The behavior of p_i under operations on tangent bundles informs the possible differentiable structures a manifold can support and the kinds of bundles that can arise as tangent or normal bundles Tangent bundle vector bundle.
Computations and applications
- Pontryagin numbers: By evaluating polynomials in the p_i on the fundamental class of a smooth closed manifold, one obtains Pontryagin numbers, which are powerful invariants in cobordism theory and the study of smooth structures. They are stable under bordism and can distinguish between manifolds that share perhaps other invariants.
- Links to topology and geometry: The p_i invariants interact with the L-classes and the χ-genus in various index-theoretic and geometric contexts, influencing how manifolds can be curved, twisted, or embedded in higher-dimensional spaces.
- Examples in geometry: In complex geometry, comparing p_i with Chern numbers provides constraints on possible complex structures, while in Riemannian geometry, curvature-based descriptions via Chern-Weil theory enable computational approaches to p_i in concrete examples Chern-Weil theory Chern class.