LOCALLY FLAT 2-KNOTS IN S2 x S2 WITH THE SAME FUNDAMENTAL GROUP

2 2 Abstract. We consider a locally flat 2-sphere in S x S representing a primi2 2 tive homology class £ , which is referred to as a 2-knot in S x S representing 2 2 (J . Then for any given primitive class ¿ , there exists a 2-knot in S x S representing { with simply-connected complement. In this paper, we consider the 2 2 classification of 2-knots in S x S whose complements have a fixed fundamental group. We show that if the complement of a 2-knot S in S x S is simply connected, then the ambient isotopy type of S is determined. In the case of nontrivial itx , however, we show that the ambient isotopy type of a 2-knot in S x 5 with nontrivial nx is not always determined by nx .