Characteristic classes

Characteristic classes
Name	Characteristic classes
Field	Topology, Differential geometry, Algebraic topology
Introduced by	Stiefel; Chern; Pontryagin
Related concepts	Vector bundle, Principal bundle, Chern–Weil theory, K-theory

Characteristic classes

Characteristic classes assign cohomology classes to fiber bundles, producing algebraic invariants that classify geometric and topological features of bundles. They connect constructions in Algebraic topology, Differential geometry, and Algebraic geometry with computations in Homotopy theory, K-theory, and representation theory, and they play roles in results of Atiyah–Singer, Hirzebruch–Riemann–Roch, and problems studied by Milnor, Stasheff, and Bott.

Introduction

Characteristic classes are cohomology classes associated functorially to principal bundles or vector bundles over a base space. Originating in work of Eduard Stiefel, Hermann Whitney, Shiing-Shen Chern, and Lev Pontryagin, they capture obstruction information and global geometric data such as orientability, complex structure, and curvature integrals. They serve as bridges between concrete geometric objects studied by Gauss, Riemann, and Chern and abstract homotopy-theoretic classification results like those of Eilenberg–MacLane, Brown, and Milnor–Moore.

Definitions and basic concepts

A characteristic class is a rule assigning to each principal G-bundle over a space X an element of H^*(X; A) for some coefficient group A, natural with respect to pullback along maps f: Y → X. Fundamental inputs include structural groups such as GL_n(R), O(n), SO(n), U(n), and Sp(n), and target cohomology theories like singular cohomology, de Rham cohomology, and generalized cohomology theories from Brown–Peterson or MU. The classification of bundles via classifying spaces such as BO, BSO, BU, and BSp yields universal characteristic classes represented by cohomology generators on these spaces; pullback along a classifying map produces the class on the specific bundle.

Primary examples and types

Primary characteristic classes include: - Stiefel–Whitney classes w_i in H^i(–; Z/2) for real bundles with structure group O(n), detecting orientability and obstructions to sections. - Chern classs c_i in integral cohomology for complex bundles with structure group U(n), central in Complex vector bundle theory and Chern–Weil theory computations. - Pontryagin classs p_i in H^{4i}(–; Z) for real bundles with structure group SO(n), related to curvature integrals and signature formulas by Hirzebruch. - The Euler class e in top-degree cohomology for oriented real bundles, appearing in Poincaré–Hopf and index computations by Atiyah and Bott. - Characteristic classes in generalized cohomology such as K-theory's Chern character and classes in elliptic cohomology connected to work by Witten.

Construction methods

Methods to construct characteristic classes include: - Classifying-space approach: use universal bundles over BO, BU, or BSpin and pull back universal cohomology classes via classifying maps from X. - Cech cocycles and obstruction theory: build classes as obstructions to trivializations or to extending sections, techniques developed by Eilenberg, Steenrod, and Postnikov. - Chern–Weil theory: for principal G-bundles with connection, apply invariant polynomials on the Lie algebra of groups like SO(n), U(n), Sp(n) to curvature forms to produce de Rham cohomology classes; foundational contributors include Chern, Weil, and Cartan. - Spectrum-level and homotopy-theoretic constructions: define classes in generalized cohomology theories using orientations and Thom isomorphisms in frameworks due to Quillen, Adams, and Boardman–Vogt.

Properties and functoriality

Characteristic classes are natural: pullback of a bundle pulls back its classes. They satisfy Whitney sum formulae expressing behavior on direct sums of bundles, multiplicative relations in the cohomology ring, and stability under stabilization by trivial bundles. Relations among classes include the total Chern class c(E⊕F)=c(E)∪c(F), relation of Stiefel–Whitney and Pontryagin classes via reduction mod 2 and complexification, and compatibility with the Thom isomorphism and Gysin maps used by Thom and Gysin. Characteristic classes detect geometric structures: vanishing of certain Stiefel–Whitney classes relates to immersions studied by Whitney and Hirsch, while relations among Pontryagin numbers constrain manifolds via Rokhlin theorem and signature formulas by Hirzebruch.

Applications and computations

Characteristic classes are used to classify bundles up to isomorphism in low dimensions, compute intersection numbers on manifolds via Hirzebruch signature theorem and Atiyah–Singer index theorem, and study cobordism and orientation questions in Complex cobordism and Spin geometry. They appear in gauge theory and instanton moduli problems studied by Donaldson and Seiberg–Witten, in anomalies in Quantum field theory studied by Witten and Alvarez-Gaumé, and in enumerative geometry via Grothendieck–Riemann–Roch. Computational tools include spectral sequences such as the Atiyah–Hirzebruch and techniques from Steenrod algebra operations studied by Cartan and Adams.

Generalizations and related theories

Generalizations extend characteristic classes to principal bundles with non-classical structure groups (e.g., Loop group bundles), to stacks and orbifolds in algebraic geometry via Deligne–Mumford techniques, and to equivariant settings using Equivariant cohomology and localization methods by Atiyah–Bott. Generalized cohomology theories produce refined classes: Morava K-theory classes, tmf classes connected to modularity and work of Hopkins–Miller, and elliptic characteristic classes tied to string structures explored by Stolz and Teichner. Interactions with Representation theory via Chern characters, with Noncommutative geometry in the work of Connes, and with categorical approaches in Higher category theory continue to expand the scope and applications of characteristic classes.

Category:Algebraic topology