# 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?

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The question seems to be a statement with a question mark after it? –  Charlie Frohman May 9 '10 at 23:34
I mean can you give a proof to say the case of general surface. –  Wolffo May 11 '10 at 14:33
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## 1 Answer

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.

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