  We elucidate magnetic effects in the skyrmion system to probe into the dense
nuclear matter. We find a deformed $\pi^0$ dipole structure of a baryon induced
by a magnetic field. We then extend our scope to stacked Skyrme crystal layers
to scrutinize phases of magnetized nuclear matter. We observe a quantized
magnetic flux and identify a phase transition from a crystalline state to a
$\pi^0$ domain wall corresponding to a topological transmutation from
$\pi_3(S^3)$ to $\pi_1(S^1)$. We establish the phase diagram, which could be
explored also in analogous systems with two-component Bose-Einstein
condensates.
