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.

 
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