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This is a followup to the post :

https://www.physicsforums.com/showthread.php?t=491211

Here Lavinia gave a couple of nice arguments showing that the complement of the solid torus the 3-sphere S^3 is a solid torus; one of which was using the Hopf fibration, taking a disk D^2 within a local trivialization of S^2, so that its lift is D^2 x S^1 , and then the complement of the disk in S^2 is itself a disk, which also lifts to a D^2 x S^1.

Now, I am trying to see if a tubular 'hood (neighborhood) (D^2 x S^1 )x S^1 of the solid torus in S^4 is again a tubular neighborhood D^2 x S^1 x S^1.

The whole thing came up from a paper which argues that a given twist extends by using a result by J.M Montesinos in which a Dehn twist extends from the embedded copy of S_3 (orientable genus-3 surface)into the whole 4-sphere S^4, if a map defined on S^1 x S^1 x S^1 induces a special type of map on the first homology.

The author starts by sugering the S^4 into a B^3 x S^1 and D^2 x S^2, after which he glues the two along their common boundary S^2 x S^1.

From what I understood, the author deforms the S_3 into a D^2 x S^1 x S^1, so that the twists are defined on the boundary S^1 x S^1 x S^1 , and then he shows that this map satisfies the conditions . But s/he seems to be saying that the complement of the tubular 'hood D^2 x S^1 x S^1 in S^4 is another D^2 x S^1 x S^1.

Is this correct?

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# Complement of a 'Hood of the Solid Torus in S^4

Can you offer guidance or do you also need help?

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