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Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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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 standa …
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