Re: fiber bundle section
- From: Michael Murray <mmurray@xxxxxxxxxxxxxxxxxxxxx>
- Date: Wed, 29 Mar 2006 13:30:07 +0000 (UTC)
On 2006-03-22 01:45:01 +1030, Einar Andreas R¿dland
What's the "simplest" fiber bundle which does not admit any global
section? If the adjective "simplest" is vague, give some conceivable
example of such a bundle.
The simplest must be the 2-to-1 map from S1 (the circle) to S1: i.e. if
S1 is identified as the complex numbers z with |z|=1, it's the map
z->z^2. A more general case is any map from Sn (n-sphere) to Pn
(projective n-space over the real numbers), which is also 2-to-1.
If you want the fibre to be connected, you can let B=S2 (sphere) and
E=unit tangent vectors at points on the sphere, with E->B mapping the
tangent vector to the base point. Then the fibers are S1 (unit vectors
in the tangent space). Since there is no everywhere nonzero tangent
vector field on S2, no global section of this exists.
Another way to think about this is it is the Z_2 bundle which is the
boundary of the Mobius band. Same thing but the geometric picture