# 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

<e.a.rodland@xxxxxxxxxxxxxx> said:

mahdiarnt wrote:

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.

Einar

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

might help.

Michael

.

**References**:**fiber bundle section***From:*mahdiarnt

**Re: fiber bundle section***From:*Einar Andreas R¿dland

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