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It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove more directly that the different definitions agree (see below). In particular it feels like I'm not understanding everything or I'm not able to see the bigger picture if I'm not able to prove this more directly.

The three definitions of the Stiefel-Whitney classes of a rank $k$ vector bundle $E$ that I'm aware of (feel free to add more) are the following:

(1) Consider the external tensor product $\hat{\otimes}$ and then the mod $2$ euler class of $\gamma \hat{\otimes} E$, where $\gamma$ is the tautological line bundle over $\mathbb{R}P^\infty$. This gives $$ e(\gamma \hat{\otimes} E)=\sum_i w_{n-i}(E)\times a^i $$ Here $a$ is the generator of $H^1(\mathbb{R}P^1)$.

(2) The obstruction theoretic definition, which I hope I don't have to state.

(3) Let $\Phi\colon H^*(B)\to H^{*+k}(E,E\setminus 0)$ denote the Thom isomorphism. Then we can define $w_i(E)=\Phi^{-1}(Sq^i(u))$, where $u\in H^k(E,E\setminus 0)$ denotes the Thom class.

I really hope that there is a direct proof of the equivalence of these 3 definitions or at least between some of these definitions.

This question can also be compared to this question. Especially the answer by Charles Rezk gives a direct proof of an equivalence of definition (1) and (3) using a neat description of the total space of $\gamma \hat{\otimes} E$ in terms of the typical involution on $S^\infty$.

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    $\begingroup$ I am not so sure how helpful Chern-Weil theory is as I'm not aware of an extension of Chern-Weil theory to produce something like Stiefel-Whitney classes. $\endgroup$ – ThorbenK Jan 4 at 14:25
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    $\begingroup$ I think it might be instructive to simply "unwind" the proof which goes through the axiomatic definition. Basically, this means recalling the proofs of naturality and all the other identities in each case (which really amount to constructions of various coboundaries etc), checking that that these definitions agree on $\mathbb R P^\infty$, and then chaining everything together. It's the part where you check that everything agrees on $\mathbb R P^\infty$ where the real geometry lives. Perhaps by reflecting on this part you can extend the argument to work directly in more generality. $\endgroup$ – Tim Campion Jan 4 at 15:10

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