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vector bundle
Definitions
- Notation: \({ \mathsf{Vect} }_r(X)\): isomorphism classes of rank \(r\) vector bundles over \(X\).
A rank \(n\) vector bundle is a ???
??? of such a bundle is a subset of \(/GL(n, k)\).
Note every rank 1 bundle is trivial: consider the Mobius strip.
Note that a vector bundle always has one section : namely, since every fiber is a vector space, you can canonically choose the 0 element in every fiber. This yields global section, the zero section.
A vector bundle \(F\to E\to B\) is trivial if \(E \cong F \times B\).
See also framed manifolds.
A rank \(n\) vector bundle is trivial iff it admits \(k\) linearly independent global sections.
The tangent bundle of a manifold is a vector bundle. Let \(M^n\) be an \(n{\hbox{-}}\)dimensional manifold. For any point \(x\in M\), the tangent bundle \({\mathbf{T}}_xM\) exists, and so we can define \begin{align*} TM = \coprod_{x\in M} {\mathbf{T}}_xM = \left\{{(x, t) \mathrel{\Big|}x\in M, t \in {\mathbf{T}}_xM}\right\} \end{align*}
Then \(TM\) is a manifold of dimension \(2n\) and there is a corresponding fiber bundle \begin{align*} {\mathbf{R}}^n \to TM \xrightarrow{\pi} M \end{align*}
given by a natural projection \(\pi:(x, t) \mapsto x\)
A circle bundle is a fiber bundle in which the fiber is isomorphic to \(S^1\) as a topological group. Consider circle bundles over a circle, which are of the form \begin{align*} S^1 \to E \xrightarrow{\pi} S^1 \end{align*}
There is a trivial bundle, when \(E = S^1 \times S^1 = T^2\), the torus:
attachments/torus-bundle 1.png
There is also a nontrivial bundle, \(E = K\), the Klein bottle:
As in the earlier example involving the orientable, \(T^2 \not\cong K\) and there are thus at least two distinct bundles of this type.
Classification
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There is an equivalence of categories between vector bundles and modules over continuous functionals: \begin{align*} {\mathsf{Bun}}({\mathbf{R}}, X)_{{\operatorname{rank}}= n} \xrightarrow{\sim} {}_{{\mathsf{Top}}(X, {\mathbf{R}})}{\mathsf{Mod}}^{{\mathrm{fg}}, \mathop{\mathrm{proj}}}_{{\operatorname{rank}}= n} .\end{align*}
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A vector bundle continuously assigns a vector space to every point of \(X\).\
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The \(k{\hbox{-}}\)dimensional vector bundles over \(X\) are equivalent to the homotopy classes of maps from \(X\) to a fixed space \([X, {\mathbf{B}}O_k]\).
- Dimension or rank???
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As with many geometric problems, classification of isomorphism classes of \(k{\hbox{-}}\)dimensional vector bundles is reduced to the computation of homotopy classes of maps.
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Studying \({\mathbf{B}}\O_k\) is very useful for this problem, it comes about by a standard construction which builds a classifying space, \({\mathbf{B}}G\), for any group \(G\).
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Complex rank 1 bundles are classified by \({\mathbf{CP}}^\infty \simeq{\mathbf{B}}{\operatorname{U}}_1({\mathbf{C}}) \simeq K({\mathbf{Z}}, 2)\).
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Universal complex vector bundle: \(\xi_{n}: E_n \rightarrow {\mathbf{B}}{\operatorname{U}}_n({\mathbf{C}})\) where \({\mathbf{B}}{\operatorname{U}}_n \cong {\operatorname{Gr}}_n({\mathbf{C}}^\infty)\) is a Grassmannian.
Todo: clean up and make precise! #todo
Unsorted
Constructions
- \(E {}^{ \vee }\otimes F \cong \mathop{\mathrm{Hom}}(E, F)\).
Exact sequences of vector bundles always split: 
Exercises
