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

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?

• The question seems to be a statement with a question mark after it? May 9, 2010 at 23:34
• I mean can you give a proof to say the case of general surface. May 11, 2010 at 14:33

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.