What makes a global section over a bundle so precious?

/ Alexander

Recently I am reading some lecture notes about Fiber Bundles & Characteristic Class, I was puzzled by the following fact:

There maybe no global sections over a fiber bundle, just a local one.

I was wondering why a global section is so precious? Why cannot one assign an element in the fiber arbitrarily ? Then after thinking whole day, I know it, that is because one may loose the continuity !

For example, we consider a \mathbb{R}^2\setminus\{0\}-bundle over \mathbb{S}^2, obtained by removing all zero vectors in its tangent bundle \mathrm{T}\mathbb{S}^2, then one can assign a tangent vector over each point whatever he/she wants, but one will never gain a smooth vector field, because of the Hairy-ball theorem.

As another example, for the 2-fold covering f: \mathbb{S}^1\longrightarrow \mathbb{S}^1, by f(z)=z^2, if we take the last circle group as \mathbb{S}^1\cong \mathbb{RP}^1=\mathbb{S}^1/\mathbb{Z}_2, it is a principal \mathbb{Z}_2-bundle, the only possible section is defined by the square root function s(z)=\sqrt{z}, however, this cannot be defined on the whole circle.

In the theory of principal G-bundle, one has the theorem :

The principal G-bundle p:E\longrightarrow B is trivial if and only if it admits a global section.

In the proof of the theorem, the continuity of the section plays a great role (but books seldom mention them!), one constructs the bundle morphism \psi: B\times G\longrightarrow E via \psi(b,g)=(b,s(b).g), where the s(b) is the assignment of b on the Lie group fiber, this is a bundle morphism if and only if that section s is continuous.

So, as we can see, the preciousness of the existence of the global sections is the preciousness of the continuity, however, we usually default the continuity in practice, or we just too lazy to check them.

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