# Differing axioms for Stiefel-Whitney class

In Milnor and Stasheff, it is taken as part of the first axiom that all Stiefel-Whitney classes of a bundle vanish in dimensions greater than the rank of the bundle. However, in other sources this is omitted. The wikipedia article makes it seem like this is actually a result of the axioms. Is there a simple proof of this (the axioms I am using are those listed here http://en.wikipedia.org/wiki/Stiefel%E2%80%93Whitney_class#Axioms)?

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## 1 Answer

It is certainly both of constructions of the Stiefel-Whitney classes that I know. The construction via the Thom Isomorphism and the Steenrod operations clearly must vanish seeing as the Thom class of the bundle will be in a dimension that is killed by the Steenrod Operation because of instability.

For the construction via universal bundles you look at the universal rank $k$-bundle $\gamma_k$, it has $w_n(\gamma_k)=0$ simply by looking at the cohomology of $BO(k)$, there isn't a polynomial generator in that dimension. The above observation coupled with naturality should give the result, but i don't know if that should count as following from the axioms.

I can flesh this out a bit later if you like.

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Thanks for the answer! I am wondering though if there is a more direct proof from the axioms (instead of going through a construction and proving uniqueness). –  Eric O. Korman Jan 9 '11 at 23:36
I understand completely. Hopefully, someone will come by with an answer. –  Sean Tilson Jan 10 '11 at 2:41
Also, it should be pointed out that it could be a mistake in the wikipedia entry... –  Sean Tilson Jan 10 '11 at 2:42