Homology of the total space of a fibre bundle

Question

Let $\pi: M^3 \to B^2$ be a compact and orientable circle bundle over a closed and orientable surface of genus $g \geq 1$. Let $c$ be a closed curve in $B$ that represents a nontrivial class in $H_1(B; \mathbb{Z})$. It it true that $\pi^{-1}(c)$ represents a nontrivial class in $H_2(M; \mathbb{Z})$?