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Does a Gysin map depend on the choice of Thom class?

Yes it does! The map in your display is (more or less) the cup product with the Thom class $\lambda_N$. So if you choose a different Thom class you'll get a different map. For example, take the ...
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What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?

Taken in the context of the introduction, I would guess that at least in part they are referring to applications of the index theorem to questions about positive scalar curvature. Before Lichnerowicz'...
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