Is there a knotted torus in 4-sphere whose complement's fundamental group is infinite cyclic ?

Question

I am reading the book 'surface in 4-space' about the unknotting conjecture (Page 97): a 2-knot (2-sphere in 4-sphere) is trival if and only if the fundamental group of the exterior is infinite cyclic.

It said that in TOP category, Freedman proved the statement is true. I don't know why it is also true for general surface. in top category?

Answer 1

I have been avoiding addressing this since almost all that I know about the question is in that book. I don't recall exactly, but I think that Kawauchi showed that a torus with the fundamental group of the complement being Z is topologically unknotted. Recent work of Hillman http://arxiv.org/pdf/1003.5408 addresses some problems of 2-knot groups, but he deals with the spherical case.

Answer 2

Now Kawauchi claims that knotted surfaces in $S^4$ whose complement has cyclic fundamental group are smoothly trivial (i.e., bound a handlebody). See Corollary 1.3.

Note: This paper currently has a gap and the unknotting conjecture is not yet settled.

Answer 3

Anthony Conway and Mark Powell give a proof that a locally flat embedded closed orientable surface in the 4-sphere whose complement has infinite cyclic fundamental group is topologically unknotted, provided that the genus is greater than or equal to 3 (the genus 0 case is a result of Freedman and Quinn as mentioned in the question).

Apart from Kawauchi's controversial work, the topological unknotting conjecture for genus 1 and 2 appears to be an open question.