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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'd suggest Guillemin and Pollack, Differential topology. You are looking for the Transversality theorem. (Also see wikipedia for the statement en.wikipedia.org/wiki/Transversality_theorem) – Sam Lisi Apr 14 '11 at 15:50

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