---
date: 2022-02-23 18:45
modification date: Sunday 3rd April 2022 17:37:16
title: "vector bundles"
aliases: [vector bundle, bundle]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #homotopy/bundles 
- Refs:
	- #todo/add-references
- Links:
	- [fiber bundle](Unsorted/fibration.md)
	- [torsor](Unsorted/torsor.md)
	- [constructing BU_n](constructing%20BU_n.md)
	- [Unsorted/vertical and horizontal subspace](Unsorted/vertical%20and%20horizontal%20subspace.md)
	- [Integration along the fiber](Unsorted/Integration%20along%20the%20fiber.md)
	- [nonabelian Hodge correspondence](Unsorted/nonabelian%20Hodge%20correspondence.md)
	- [[parabolic bundle]]
	- [[Atiyah bundle]]
	- [isomonodromic deformation](Unsorted/isomonodromic%20deformation.md)
	- [Higgs bundle](Unsorted/Higgs%20bundle.md)

---

# vector bundle

## Definitions 

- Notation: $\Vect_r(X)$: isomorphism classes of rank $r$ vector bundles over $X$.


:::{.definition title="Rank of a vector bundle"}
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.

:::{.remark}
Note that a vector bundle always has one [section](section) : namely, since every fiber is a vector space, you can canonically choose the 0 element in every fiber.
This yields [global section](global%20section), the [zero  section](zero%20%20section).
:::

:::{.definition title="Trivial bundle"}
A vector bundle $F\to E\to B$ is **trivial** if $E \cong F \cross B$.
:::

See also [framed](framed.md) manifolds.

:::{.proposition title="Trivial iff linearly independent sections"}
A rank $n$ vector bundle is trivial iff it admits $k$ linearly independent global sections.
:::

:::{.example title="?"}
The [tangent bundle](tangent%20bundle.md) of a manifold is a vector bundle. Let $M^n$ be an $n\dash$dimensional manifold. For any point $x\in M$, the tangent  bundle $\T_xM$ exists, and so we can define
$$
TM = \coprod_{x\in M} \T_xM = \theset{(x, t) \mid x\in M, t \in \T_xM}
$$

Then $TM$ is a manifold of dimension $2n$ and there is a corresponding fiber bundle
$$
\RR^n \to TM \mapsvia{\pi} M
$$

given by a natural projection $\pi:(x, t) \mapsto x$
:::

:::{.example title="?"}
A circle bundle is a [fiber  bundle](fiber%20%20bundle) in which the fiber is isomorphic to $S^1$ as a topological group. Consider circle bundles over a circle, which are of the form
$$
S^1 \to E \mapsvia{\pi} S^1
$$

There is a trivial bundle, when $E = S^1 \cross S^1 = T^2$, the torus:

![attachments/torus-bundle 1.png](attachments/torus-bundle%201.png)

There is also a nontrivial bundle, $E = K$, the Klein bottle:

![klein-bottle.png](attachments/klein-bottle.png)

As in the earlier example involving the [orientable](orientable.md), $T^2 \not\cong K$ and there are thus at least two distinct bundles of this type.
:::


# Classification

-  There is an equivalence of categories between vector bundles and modules over continuous functionals:
$$
\Bun(\RR, X)_{\rk = n} \mapsvia{\sim} \modsleft{\Top(X, \RR)}^{\fg, \proj}_{\rk = n}
.$$ 

- A vector bundle continuously assigns a vector space to every point of $X$.  
- The $k\dash$dimensional vector bundles over $X$ are equivalent to the homotopy classes of maps from $X$ to a fixed space $[X, \B O_k]$. 
	- Dimension or rank???
- As with many geometric problems, classification of isomorphism classes of $k\dash$dimensional vector bundles is reduced to the computation of homotopy classes of maps. 
- Studying $\B\O_k$ is very useful for this problem, it comes about by a standard construction which builds a [classifying space](classifying%20space.md),  $\B G$, for any group $G$.

- Complex rank 1 bundles are classified by $\CP^\infty \homotopic \B \U_1(\CC)  \homotopic K(\ZZ, 2)$.
- Universal complex vector bundle:
$\xi_{n}: E_n \rightarrow \B \U_n(\CC)$
	where $\B \U_n \cong \Gr_n(\CC^\infty)$ is a [Grassmannian](Grassmannian.md).
	
	
Todo: clean up and make precise! #todo 


# Unsorted

![](attachments/Pasted%20image%2020210613122600.png)

![](attachments/Pasted%20image%2020210613122600.png)

![](attachments/Pasted%20image%2020210613122620.png)

# Constructions
![](attachments/Pasted%20image%2020220403173520.png)

![](attachments/Pasted%20image%2020210613122630.png)

![](attachments/Pasted%20image%2020210613122725.png)

- $E\dual \tensor F \cong \Hom(E, F)$.


Exact sequences of vector bundles always split:
![](attachments/Pasted%20image%2020220403173715.png)

# Exercises

![](attachments/Pasted%20image%2020220403173615.png)