Let $M$ be the underlying topological space of a K3 surface. Is there a Serre fibration $M\to B$ where $B$ is a CW complex of dimension $0<d<4$?

Let $M$ be the underlying topological manifold of a smooth complex projective surface. Assume $\pi_1(M)=\{0\}$ and $\pi_2(M)\neq \mathbb{Z}^2$. 

Is there a Serre fibration $M\to B$ where $B$ is a CW complex of dimension $0<d<4$?

Let $M$ be the underlying topological space of a K3 surface. Is there a Serre fibration $M\to B$ where $B$ is a finite CW complex of dimension $0<d<4$?

Let $M$ be the underlying topological space of a K3 surface. Is there a Serre fibration $M\to B$ where $B$ is a CW complex of dimension $0<d<4$?