First-year graduateGeometry & TopologyLesson available

Vector bundles and characteristic classes

Families of vector spaces parametrised by a space: vector bundles, the tangent bundle, clutching and classifying spaces, Stiefel–Whitney, Chern, Euler and Pontryagin classes, Chern–Weil theory linking curvature to topology, and the Gauss–Bonnet–Chern theorem.

math.geom.vector-bundles-characteristic-classes

Open lesson Show everything needed to reach this topic

Builds on

Algebraic topology Smooth manifolds Differential forms and Stokes' theorem

Required prerequisites, all the way down: 49

You will learn to

Define vector bundles, sections and pullbacks, and recognise the tangent bundle and the Möbius line bundle.
Define Stiefel–Whitney and Chern classes axiomatically and use them to obstruct the existence of sections and embeddings.
Use Chern–Weil theory to express characteristic classes through curvature and state the Gauss–Bonnet–Chern theorem.

Definition

A real vector bundle of rank k over B is a map p : E → B whose fibres are k-dimensional vector spaces, locally isomorphic to B × ℝᵏ.
A bundle is trivial if it is isomorphic to the product bundle B × ℝᵏ.
Characteristic classes assign to each bundle cohomology classes of the base, natural under pullback.

Theorem

The Möbius line bundle over the circle is non-trivial: its first Stiefel–Whitney class is non-zero.
Rank-k complex vector bundles over a paracompact space B correspond to homotopy classes of maps B → BU(k).
Gauss–Bonnet–Chern: for a closed oriented even-dimensional Riemannian manifold, the integral of the Pfaffian of the curvature equals (2π)^{n/2} times the Euler characteristic (with the standard normalisation).

Common misconceptions

Thinking every bundle with trivial fibres and contractible-looking pieces is trivial; local triviality does not imply global triviality.
Believing the tangent bundle of a sphere is trivial whenever the sphere is 'round'; TSⁿ is trivial only for n = 0, 1, 3, 7.

Leads to

Complex and Kähler geometry; Hodge theory Stable homotopy theory and generalised cohomology Gauge theory and the geometry of physics

Optional discoveries

Gauge theory and the geometry of physics A gauge field is a connection on a principal bundle; its curvature is the field strength, and characteristic classes of the bundle give topological charges such as instanton number.
Stable homotopy theory and generalised cohomology Topological K-theory, built from vector bundles, is the first example of a generalised cohomology theory represented by a spectrum.

Learning pathways

Geometry and topology