# Transversality of the section

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?

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I'll answer the easy one: $s$ and $s'$ are always isotopic, since you can consider the isotopy $s_t = t\cdot s'$ (using the linear structure on each fiber, so $t\cdot s'$ is well-defined). –  Marco Golla Apr 25 '11 at 14:50

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 $A$.