Let $X\_1$ and $X\_2$ be two closed spin$^c$ manifolds that are bordant via a spin$^c$ manifold-with-boundary $W$.


Let $Z$ be a closed spin$^c$ manifold with $\dim Z=\dim X\_1$ mod $2$. Let
$$f\_1:X\_1\to Z,\qquad f\_2:X\_2\to Z,\qquad F:W\to Z$$
be smooth maps such that $F|\_{X\_1}=f\_1$ and $F|\_{X\_2}=f\_2$. We can associate to $f\_1$ and $f\_2$ two wrong-way (or Gysin) maps in $K$-theory:


$$f\_{1!}:K^0(X\_1)\to K^0(Z),$$
$$f\_{2!}:K^0(X\_2)\to K^0(Z).$$


Let $E\_1\to X\_1$ and $E\_2\to X\_2$ be two $\mathbb{C}$-vector bundles such that there exists a vector bundle $\Omega\to W$ satisfying $\Omega|\_{X\_1}\cong E\_1$ and $\Omega|\_{X\_2}\cong E\_2$. Let $[E\_i]\in K^0(X\_i)$ denote the $K$-theory classes defined by $E\_i$.


**Question:** Is it true that $f\_{1!}[E\_1]=f\_{2!}[E\_2]\in K^0(Z)$?


*Added after:* I would be most interested in an approach not directly using Poincare duality for K-theory/K-homology.



Does the Gysin map in $K$-theory respect bordism?
Let $N^n=\partial M^{n+1}$, $E\in K^\bullet(M)$ and $f:M\to X$


Choose a smooth embedding $i:X\to \mathbb{R}^N,N>>1$,
denote by $\chi$ the normal bundle of $X$ and by $\mu$ the normal bundle of $M$ after suitable small deformation of $i\circ f$.


Let $\nu=\mu|\_N$ and $\eta$ be the normal bundle of $N\subset M$ (which is trivial and one-dimensional)


By considering tubular neighborhoods we get the natural map:


$t:Th\_\chi X\to Th\_{\nu+\eta}N$, where $Th$ denotes a Thom space.


After applying the Thom isomorphism $th$ on $K^\bullet$ we obtain the definition of a Gysin map (going in "right-way" on a $Th$'s). So for $f\_!(E|\_N)=0$ it's sufficient to prove that $t^\* th\_{\nu+\eta}(E|\_N)=0$


Actually $t^\*$ is passing through a connecting homomorphism.
Namely, there is a commutative diagram:


$\begin{matrix}
Th\_{\chi}X&\to& Th\_{\mu}M/Th\_\nu N&\\
\downarrow{t}&\swarrow{\sigma}&\downarrow{\Sigma}&\\
Th\_{\nu+\eta}N&\xrightarrow{\sim}& \Sigma Th\_{\nu}N&\\
\end{matrix}$


The top arrow comes from the tubular neighborhoods.


The horizontal isomorphism comes from triviality of $\eta$, while suspension $\Sigma$ from Puppe cofiber sequence:


$Th\_\nu N\to Th\_\mu M\to Th\_\mu M/Th\_\nu N\xrightarrow{\Sigma} \Sigma Th\_{\nu}N$


The map $\sigma$ explains commutativity and is coming from:


$Th\_\mu M/Th\_\nu N\sim Th\_\mu M/Th\_\mu (N\times [0,\varepsilon))\to$
$Th\_\mu (N\times(-\varepsilon,\varepsilon))/Th\_\mu (N\times [0,\varepsilon))\to Th\_{\nu+\eta}N$
where $N\times [0,\varepsilon)\subset M$ is a collar of $N$.


Finally, $\Sigma^\*$ is the connecting homorphism and it follows that $\Sigma^\* th\_{\nu}(F)=0$ for all $F\in Im( K^\bullet(M)\to K^\bullet(N))$, so $t^\* th\_{\nu+\eta}(E|\_N)=0$


