nLab
universal vector bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

In topology a universal vector bundle of some rank nn is a vector bundle ζ nBGL(n)\zeta_n \to B GL(n) (over a base space to be called a classifying space) such that every other vector bundle EXE \to X of rank nn over a suitably nice topological space (paracompact topological space) arises as the pullback bundle Ef *ζ nE \simeq f^\ast \zeta_n of the universal bundle, along some morphisms (continuous function) f:XBGL(n)f \colon X \to B GL(n) which is unique up to homotopy:

E ζ n (pb) X f BGL(n). \array{ E &\longrightarrow& \zeta_n \\ \downarrow &(pb)& \downarrow \\ X &\underset{f}{\longrightarrow}& B GL(n) } \,.

The universal real vector ζ n\zeta_n of rank nn is the vector bundle which is associated to the universal principal bundle EGL(n)BGL(n)E GL(n) \to B GL(n) (with structure group the general linear group) over the given classifying space, equivalently to EO(n)BO(n)E O(n) \to B O(n):

ζ n(EO(n))×O(n) n. \zeta_n \coloneqq (E O(n))\underset{O(n)}{\times} \mathbb{R}^n \,.

Similarly for complex vector bundles

ζ n (EU(n))×U(n) 2n. \zeta^{\mathbb{C}}_n \coloneqq (E U(n))\underset{U(n)}{\times} \mathbb{R}^{2n} \,.

etc.

Constructions

Via Grassmannians and Stiefel manifolds

For n,kn, k \in \mathbb{N}, and nkn \leq k, there is the Grassmannian manifold given as the coset topological space

Gr n(k)O(k)/(O(n)×O(kn)). Gr_n(k) \coloneqq O(k)/(O(n)\times O(k-n)) \,.

Similarly, the Stiefel manifold is the coset

V n(k)O(k)/O(n). V_n(k) \coloneqq O(k)/O(n) \,.

The quotient projection

V kn(k)Gr n(k) V_{k-n}(k)\longrightarrow Gr_n(k)

is an O(n)O(n)-principal bundle, with associated bundle V n(k)× O(n) nV_n(k)\times_{O(n)} \mathbb{R}^n a vector bundle of rank nn. In the limit (colimit) that kk \to \infty is this gives a presentation of the O(n)O(n)-universal principal bundle and of the universal vector bundle of rank nn, respectively.. The base space Gr n() wheBO(n)Gr_n(\infty)\simeq_{whe} B O(n) is the classifying space for O(n)O(n)-principal bundles and rank nn vector bundles.

Examples

References

Textbook accounts include

Revised on May 30, 2017 15:10:36 by Anonymous (128.84.125.207)