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Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom isomorphism $$K^{\bullet}(X)\rightarrow \tilde{K}^{\bullet}(\text{Th}(N))$$ in topological $K$-theory. When $X,Y$ are compact, we can then define a Gysin-Thom map $f_*:K^{\bullet}(X)\rightarrow K^{\bullet}(Y)$ by tubular neighbourhood theorem.

Does the Gysin map depend on our choice of the Thom class (aka. the $\textsf{spin}^c$ structure) on $N$)?

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom isomorphism $$K^{\bullet}(X)\rightarrow \tilde{K}^{\bullet}(\text{Th}(N))$$ in topological $K$-theory. When $X,Y$ are compact, we can then define a Gysin-Thom map $f_*:K^{\bullet}(X)\rightarrow K^{\bullet}(Y)$ by tubular neighbourhood theorem.

Does the Gysin map depend on our choice of the Thom class (aka. the $\textsf{spin}^c$ structure on $N$)?

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom isomorphism $$K^{\bullet}(X)\rightarrow \tilde{K}^{\bullet}(\text{Th}(N))$$ in topological $K$-theory. When $X,Y$ are compact, we can then define a Gysin-Thom map $f_*:K^{\bullet}(X)\rightarrow K^{\bullet}(Y)$ by tubular neighbourhood theorem.

Does the Gysin map depend on our choice of the Thom class (aka. the $\textsf{spin}^c$ structure) on $N$?

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user484289
user484289

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom isomorphism $$K^{\bullet}(X)\rightarrow K^{\bullet}(N)$$$$K^{\bullet}(X)\rightarrow \tilde{K}^{\bullet}(\text{Th}(N))$$ in topological $K$-theory. WeWhen $X,Y$ are compact, we can then define a Gysin-Thom map $f_*:K^{\bullet}(X)\rightarrow K^{\bullet}(Y)$ by tubular neighbourhood theorem.

Does the Gysin map depend on our choice of the Thom class (aka. the $\textsf{spin}^c$ structure on $N$)?

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom isomorphism $$K^{\bullet}(X)\rightarrow K^{\bullet}(N)$$ in topological $K$-theory. We can then define a Gysin-Thom map $f_*:K^{\bullet}(X)\rightarrow K^{\bullet}(Y)$ by tubular neighbourhood theorem.

Does the Gysin map depend on our choice of the Thom class (aka. the $\textsf{spin}^c$ structure on $N$)?

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom isomorphism $$K^{\bullet}(X)\rightarrow \tilde{K}^{\bullet}(\text{Th}(N))$$ in topological $K$-theory. When $X,Y$ are compact, we can then define a Gysin-Thom map $f_*:K^{\bullet}(X)\rightarrow K^{\bullet}(Y)$ by tubular neighbourhood theorem.

Does the Gysin map depend on our choice of the Thom class (aka. the $\textsf{spin}^c$ structure on $N$)?

Source Link
user484289
user484289

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

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom isomorphism $$K^{\bullet}(X)\rightarrow K^{\bullet}(N)$$ in topological $K$-theory. We can then define a Gysin-Thom map $f_*:K^{\bullet}(X)\rightarrow K^{\bullet}(Y)$ by tubular neighbourhood theorem.

Does the Gysin map depend on our choice of the Thom class (aka. the $\textsf{spin}^c$ structure on $N$)?