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

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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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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.

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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.

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