Let $E\to X$ be an oriented real $n$-plane bundle. Then the Euler class $e(E)\in H^n(X;\Bbb Z)$ can be defined, using Thom isomorphism (https://en.wikipedia.org/wiki/Euler_class#Formal_definition). I am curious about its properties, given here: https://en.wikipedia.org/wiki/Euler_class#Properties.
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As Chern classes or Stiefel-Whitney classes, does the four properties (functoriality, Whitney sum formula, normalization, and orientation) uniquely characterize the Euler class?
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It is also written that if $X$ is an oriented smooth $d$-manifold and $\sigma:X\to E$ is a smooth section that intersects the zero section transversally, then $e(E)$ is the Poincare dual of the class in $H_{d-r}(X;\Bbb Z)$ represented by the zero locus of $\sigma$. How can this be proved? Is there a reference of a proof for this statement?
Best Answer
The standard reference for characteristic classes is the book by Milnor and Stasheff "Characteristic Classes", and I recommend reading it.
Your second question is also answerd in these seminar notes ( Theorem 3.2) by Matthias Görg.
The answer to your first question is no. They don't make sure we just have $e(E)=0$ for all bundles.
Add a fifth axiom, saying something like $e(\gamma)[\mathbb{CP}^1] = -1 $ , where $\gamma \to \mathbb{CP}^1$ is the tautological bundle ( as an oriented $2$-plane bundle) and you can proove uniqueness, using a real splitting principle.
Edit: As pointed out in the comments, the uniqueness part is not working the way I thought. I posted this as a new question here.