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I've been reading Hatcher's book on vector bundles and I'm just getting into the section of Steifel-Whitney numbers. Naturally, I'm interested in which sequences of such numbers are realizable, and Googling revealed the following MO thread. Unfortunately, the answers there all seem a little opaque to me. Does anyone know whether the following admits are more understandable answer: If the SW class of a bundle is non-trivial, are there any restrictions on the first $i$ such that the $i$th S-W class is nonzero?

Probably a stupid question, but they tend to help me learn.

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