Mathematics.

algebraic topology

Vector Bundles

Topology100 minDifficulty8 out of 10

You should know: fiber bundles

Overview

A vector bundle is a fiber bundle whose fiber is a vector space and whose structure group is the general linear group. Vector bundles arise naturally in differential geometry (tangent and cotangent bundles), algebraic topology (the tautological bundle over projective space), and physics (gauge theory). Their study leads to K-theory, characteristic classes, and index theorems.

Intuition

Attach a vector space to each point of a base space B in a smoothly varying way. The tangent bundle of a surface is the canonical example: at each point of the surface, you attach the tangent plane. Globally these planes may twist around each other, and characteristic classes measure this twisting.

Formal Definition

Definition

A rank-k real vector bundle over B is a fiber bundle pi: E -> B with fiber R^k and structure group GL(k, R), such that the transition functions g_{alpha beta}: U_alpha ∩ U_beta -> GL(k, R) are continuous and the local trivializations preserve the vector space structure on each fiber. A section of E is a continuous map s: B -> E with pi ∘ s = id_B.

Rk↪E→πB\mathbb{R}^k \hookrightarrow E \xrightarrow{\pi} B
Rank-k vector bundle
gαβ:Uα∩Uβ→GL(k,R)g_{\alpha\beta}: U_\alpha\cap U_\beta \to GL(k,\mathbb{R})
Transition functions
E∣Uα≅Uα×RkE|_{U_\alpha} \cong U_\alpha \times \mathbb{R}^k
Local trivialization

Properties

Whitney Sum

GivenvectorbundlesE→BandF→B,theirWhitneysumE⊕F→BisthebundlewithfiberEx⊕Fxateachx.Given vector bundles E \to B and F \to B, their Whitney sum E\oplus F \to B is the bundle with fiber E_x\oplus F_x at each x.

Tensor Product

GivenvectorbundlesEandFoverB,theirtensorproductE⊗F→BhasfiberEx⊗Fxateachx.Given vector bundles E and F over B, their tensor product E\otimes F \to B has fiber E_x\otimes F_x at each x.

Dual Bundle

ThedualbundleE∗hasfiber(Ex)∗ateachx.The dual bundle E^* has fiber (E_x)^* at each x.

Theorems

Theorem 1: Hairy Ball Theorem
ThetangentbundleTS2hasnonowhere−vanishingsection;equivalently,S2doesnotadmitacontinuousnonzerovectorfield.The tangent bundle TS^2 has no nowhere-vanishing section; equivalently, S^2 does not admit a continuous nonzero vector field.
Theorem 2: Classification by Homotopy Classes
Isomorphismclassesofrank−krealvectorbundlesoveraCWcomplexBareinnaturalbijectionwithhomotopyclasses[B,BO(k)],whereBO(k)istheclassifyingspace.Isomorphism classes of rank-k real vector bundles over a CW complex B are in natural bijection with homotopy classes [B, BO(k)], where BO(k) is the classifying space.
Theorem 3: Splitting Principle
ForanyvectorbundleE→B,thereexistsaspaceF(E)andamapp:F(E)→Bsuchthatp∗isinjectiveoncohomologyandp∗Esplitsasasumoflinebundles.For any vector bundle E \to B, there exists a space F(E) and a map p: F(E)\to B such that p^* is injective on cohomology and p^*E splits as a sum of line bundles.

Worked Examples

  1. For S^1: parametrize by angle theta. The vector d/dtheta is a nowhere-vanishing global section, so TS^1 ≅ S^1 × R.

  2. For S^2: suppose TS^2 were trivial, i.e., S^2 × R^2 ≅ TS^2. Then there would exist two linearly independent everywhere-nonzero vector fields on S^2.

  3. But even one nonzero vector field would contradict the Hairy Ball Theorem, which says every continuous vector field on S^2 vanishes somewhere.

  4. Thus TS^2 is nontrivial. (Its Euler class is 2 times the fundamental class, confirming nontriviality via characteristic classes.)

Answer: TS^1 is trivial (admits a global frame); TS^2 is nontrivial (the hairy ball theorem forbids a nonzero global section).

Practice Problems

Difficulty 6/10

Describe the cotangent bundle T^*S^2 and determine whether it is trivial.

Difficulty 8/10

Prove that every vector bundle over a contractible space is trivial.

Difficulty 7/10

Let E -> B be a rank-k vector bundle and F -> B a rank-m vector bundle. What is the rank of E ⊗ F?

Common Mistakes

Common Mistake

A trivial bundle must have all transition functions equal to the identity matrix.

A bundle is trivial if transition functions can be chosen to be constant (equal to the identity), not that they are automatically so in every trivialization.

Common Mistake

The zero section is a nonzero section of a vector bundle.

The zero section assigns the zero vector to each point. A nowhere-vanishing section assigns a nonzero vector. These are different notions.

Quiz

What is the rank of the tangent bundle of a smooth n-manifold?
Which space classifies rank-k real vector bundles?
The Hairy Ball Theorem says that which sphere has no nonzero continuous vector field?

Summary

  • A vector bundle is a fiber bundle with vector space fibers and GL(k) structure group, generalizing tangent and cotangent bundles.
  • Operations include Whitney sum, tensor product, dual, and exterior powers, all producing new bundles.
  • A bundle is trivial iff it admits k linearly independent global sections; TS^2 is nontrivial by the Hairy Ball Theorem.
  • The classification theorem: rank-k bundles over B correspond bijectively to homotopy classes [B, BO(k)].
  • Characteristic classes (Stiefel-Whitney, Chern, Pontryagin) are cohomology classes that measure the twisting of vector bundles.

References

  1. BookMilnor, J. & Stasheff, J. — Characteristic Classes. Princeton University Press, 1974.
  2. WebsiteWikipedia — Vector bundle