section of a fiber bundle

Question

I heard in class that not every fiber bundle admits a section. I am not sure why this is true, you can always pick a point on a fiber and follow it through as you glue local trivializations then you get a section, isn't this right?

Answer 1

The "point" (pun intended) is that there is no canonical identification between the fibers over various points in the base. So, it is not possible to simply "pick a point on a fiber and follow it through". In the important case of a vector bundle, there is always a section - the zero section - because the zero vector is canonical. In general, however, a fiber bundle may not admit a section - for example, the $\mathbb{Z}/2$-bundle $S^n\to \mathbb{RP}^n$ does not admit a section. (Why?)

Hope this helps!

Answer 2

Let $M$ be a Möbius strip, $C\subset M$ a circle running through the center, and consider the fiber bundle $M\setminus C \to C$ given by some projection.

What happens if we try to construct a section in the way you describe? For each $c\in C$, we need to pick a point either above or below $C$.

But as you can easily verify with some paper, $M\setminus C$ is connected! So if we "follow it through", we will find that there is no consistent choice of side.

Answer 3

Here is a counterexample from differential topology.

Take the base space to be the two dimensional sphere $S^2$. Consider the tangent bundle $TS^2$, which of course has a section, e.g. the zero section. Now simply remove that section: let $A$ be obtained from $TS^2$ by removing the zero vector over each point of $S^2$. Then the bundle $A \mapsto S^2$ has no section: this is just a restatement of the "hairy ball" theorem from differential topology, which says that $S^2$ has no nonzero vector field.

Answer 4

For general fiber bundles it is not true as the other answers have pointed out. As you put some more structure on the bundle then it might or might not be true.

For example:

(i) Every vector bundle admits the zero section.

(ii) A principal bundle admits a (global) section if and only if is trivial.

Maybe your teacher had a more specialized class of bundles in mind when he stated that.