The Unknotting Problem for Surfaces in the 4-sphere

We determine explicit integers $g_1$ and $g_2$ such that if $g$ is at least $g_1$ (resp. $g_2$) then there is no algorithm to decide whether or not a closed, orientable, PL locally flat surface in $S^4$ of genus $g$ is PL (resp. TOP) unknotted. We also give analogous results for non-orientable surfaces, and for 2-spheres in connected sums of $S^2 x S^2$.