Non-vanishing section on compact manifolds

Question

Now if we have a compact smooth manifold M and a rank k vector bundle on it. Then I want to find a non-vanishing smooth section on M if $k>dim M$. But I have met some difficulties: The main idea is similar to the case when we proof the weak Whitney embedding theorem. Suppose
$\eta:M\rightarrow E$ Is the zero section, since M is compact, it is also an embedding, which we denote the submanifold by $S=\eta(M)$, then we want to show that there is section $\sigma:M\rightarrow E$, whose image intersects with S empty. It seems to me that the amount of this kind of $\sigma$ Is very large due to dimensional reasons, but I just don’t know how to extract one of them? How can I make it?

Answer 1

I actually feel that this is a very important structural theorem about vector bundles that is mysteriously hard to find in any introductory textbook.

The main theorem that I'll use can be found in Bredon's "Topology and Geometry" under number II.15.3:

Let $E\overset{\pi}{\to}M$ be a smooth vector bundle over a smooth manifold $M$. Let $M'$ be a smooth manifold and $f:M'\to E$ a smooth map. Then there is a smooth section $\sigma:M\to E$ that can be chosen arbitrarily close to the zero section, such that $\sigma\pitchfork f$.

Here $\pitchfork$ denotes transversality of maps, i.e. the images of the differentials of the two maps sum up to the entire tangent space in the codomain at every point in the intersection of their images.

Now, if $E\overset{\pi}{\to}M^m$ has rank $k$ and $m=\mathrm{dim} M$, then we have three cases:

1. $k>m$. Then by the above theorem, there is a section $\sigma$ that is transverse to the zero section $E_0$, denoted $\sigma_0:M\to E$. By dimension counting, this implies that $\mathrm{im}(\sigma)\cap E_0=\varnothing$. Then, by induction, $E$ contains a rank $k-m$ trivial subbundle. For this reason, vector bundles of rank higher than the dimension of the base are not very interesting from the point of view of any classification of bundles (e.g. K-theory).
2. $k=m$. In this case the "generic" section of $E$ is transverse to the zero section and by dimension counting has only isolated zeros. This is the case e.g. for sections of tangent bundles, or complex line bundles over Riemann surfaces, and allows one to define the degree of a section.
3. $k<m$. In this case the "generic" section of $E$ is transverse to the zero section and turns to zero along a submanifold of $M$ of dimension $m-k$.