# Bundles on spheres

## Clutching functions

A convenient way of constructing vector bundles over spheres ${\displaystyle S^{k}}$ is by using clutching functions. We start by decomposing ${\displaystyle S^{k}}$ as the disjoint union of two disks ${\displaystyle D_{+}^{k}}$ and ${\displaystyle D_{-}^{k}}$. Each disc is contractible and so any vector bundle on each piece is trivial by Fact 4.1. We then glue along the boundaries by identifying ${\displaystyle \partial D_{+}^{k}}$ with ${\displaystyle \partial D_{-}^{k}}$, whilst applying a map called a clutching function (think of engaging gears in machinery) ${\displaystyle f:S^{k-1}\rightarrow GL_{r}(\mathbb {C} )}$ to the vectors in the bundle. We can thicken the intersection slightly to ${\displaystyle S^{k-1}\times (-\epsilon ,\epsilon )}$ to make things smooth.

Remark

Homotopic clutching functions ${\displaystyle f}$, ${\displaystyle g}$ give isomorphic smooth complex vector bundles ${\displaystyle E_{f}}$, ${\displaystyle E_{g}}$.

Theorem 4.1

There is the following bijection between homotopy classes of maps and rank ${\displaystyle r}$ complex vector bundles over ${\displaystyle S^{k}}$:

${\displaystyle [S^{k-1},GL_{r}(\mathbb {C} )]\leftrightarrow V_{\mathbb {C} }^{r}(S^{k})}$

Proof

See Hatcher, Prop 1.11

Corollary 4.1

Every complex vector bundle over ${\displaystyle S^{1}}$ is trivial.

Proof

This is equivalent to saying that ${\displaystyle GL_{r}(\mathbb {C} )}$ is path-connected. By applying elementary row operations, we can diagonalise any matrix ${\displaystyle GL_{r}(\mathbb {C} )}$. We construct a path to the diagonal matrix by applying an appropriate sequence of row operations with a factor of ${\displaystyle \lambda }$ in front of each, then running ${\displaystyle \lambda }$ continuously from 0 to 1. Diagonal matrices in ${\displaystyle GL_{r}(\mathbb {C} )}$ are homeomorphic to ${\displaystyle r}$ copies of ${\displaystyle \mathbb {C} -\{0\}}$ which is path-connected.

Example 4.1

For the line bundle on ${\displaystyle S^{2}}$ we have: ${\displaystyle \pi _{1}(GL_{1}(\mathbb {C} ))=\pi _{1}(\mathbb {C} ^{*})=\pi _{1}(\mathbb {Z} )\cong \mathbb {Z} }$ without wishing to jump the gun, we will soon see that the cohomology class of the generator of this group is the first Chern class of the line bundle.

Exercise 4.3

Show that a line bundle ${\displaystyle L}$ is trivial ${\displaystyle \iff }$${\displaystyle L}$ has a non-vanishing section. Extension: Show that a rank ${\displaystyle r}$ vector bundle ${\displaystyle E}$ is trivial ${\displaystyle \iff }$${\displaystyle E}$ has ${\displaystyle r}$ sections ${\displaystyle s_{1},\ldots ,s_{r}}$ such that the vectors ${\displaystyle s_{1}(b),\ldots ,s_{r}(b)}$ are linearly independent in each fibre ${\displaystyle p^{-1}(b)}$

Exercise 4.4

Research and understand the Hopf fibrations.