Question

Let $M^n$ be a differentiable manifold and $\pi\colon E\to M$ is $n$-dimensional vector bundle over $M$.

We have a zero section $s\colon M\to E$ of $\pi$.

How can I make a section $s'$ which is trnasversal to $s$? (i.e., $s'$ vanishes $s$ finitely many times.)

(In some text, it seems even possible to make $s$ and $s'$ are isotopic.)

I need this to interpret the euler class of $\pi$, $\chi(\pi)$ as an algebraic intersection number of $s$ and $s'$.

Are there anybody who can give me any references?

Answer 1

This is a straightforward application of transversality theorem, which roughly speaking states that we can make a map transverse to a submanifold with an arbitrary small perturbation. It is a consequence of Morse-Sard theorem.

The statement you need is the following:

Theorem. Let $A$, $B$ be $\mathcal{C}^r$-submanifolds of $M$, $1 \leq r \leq \infty$. Then every neighborhood of the inclusion $i_B \colon B \to M$ in $\mathcal{C}^r(B, M)$ contains an embedding which is transverse to $A$.

For a proof, see [Hirsch, Differential Topology, Thm. 2.4 pag. 78].