Knot groups in<i>S</i>⁴with nontrivial homology

Abstract

In this paper we exhibit smooth 2-manifolds F 2 in the 4-sphere S 4 having the property that the second homology of the group 1 (S i -F 2 ) is nontrivial. In particular, we obtain tori for which H 2 { x )=Z 2 and, by forming connected sums, surfaces of genus n for which i^fe) is the direct sum of n copies of Z 2 . Corollaries include: (1) There are knotted surfaces in S 4 that cannot be constructed by forming connected sums of unknotted surfaces and knotted 2-spheres. (2) The class of groups that occur as knot groups of surfaces in S 4 is not contained in the class of high dimensional knot groups of S n in S n+2 .