# Bundles on spheres

## Clutching functions[edit | edit source]

A convenient way of constructing vector bundles over spheres is by using clutching functions. We start by decomposing as the disjoint union of two disks and . 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 with , whilst applying a map called a **clutching function** (think of engaging gears in machinery) to the vectors in the bundle. We can thicken the intersection slightly to to make things smooth.

**Remark**

Homotopic clutching functions , give isomorphic smooth complex vector bundles , .

**Theorem 4.1**

There is the following bijection between homotopy classes of maps and rank complex vector bundles over :

*Proof*

See Hatcher, Prop 1.11

**Corollary 4.1**

Every complex vector bundle over is trivial.

*Proof*

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

**Example 4.1**

For the line bundle on we have: 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 is trivial has a non-vanishing section. Extension: Show that a rank vector bundle is trivial has sections such that the vectors are linearly independent in each fibre

**Exercise 4.4**

Research and understand the Hopf fibrations.