Let $f: M \to N$ be a symplectomorphism and $\omega_N$ a symplectic form on $N$. I want to calculate the symplectic chern class $c_1(TM, f^* \omega_N)$.
Assume that I know a symplectic form $\omega_M$ on $M$ with $[\omega_M] = [f^*\omega_N]\in H^2(M;\mathbb R)$ and $c_1(TM, \omega_M)=0$.
Can I conclude that $c_1(TM, f^*\omega_N)=0$?
Edit: Actually, I know more than that $[\omega_M] = [f^* \omega_N]$. I know that $f$ is homotopic to a symplectomorphism $g$ with $g^* \omega_N = \omega_M.$
Edit2: $f$ is a covering map