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.