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.