स्नातकोत्तर प्रथम वर्षGeometry & Topologyपाठ उपलब्ध

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

पाठ खोलें इस विषय तक पहुँचने के लिए आवश्यक सब कुछ दिखाएँ

इस पर आधारित

Algebraic topology Smooth manifolds Differential forms and Stokes' theorem

49 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)

आप सीखेंगे

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.

परिभाषा

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.

प्रमेय

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).

आम भ्रांतियाँ

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.

आगे ले जाता है

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

वैकल्पिक खोजें

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.

अधिगम पथ

Geometry and topology

आगे पढ़ने के लिए

John Milnor and James D. Stasheff, Characteristic Classes (1974)
Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology (1982)