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

[Bacle]

Hi, All:

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?

 

[jvicens]

Hello,

Thank you for bringing this interesting problem to our attention. After reading through the original post and your follow-up, I believe that the author's statement about the complement of the tubular neighborhood being another D^2 x S^1 x S^1 is incorrect.

Firstly, let's clarify what a tubular neighborhood is. In general, a tubular neighborhood is a neighborhood of a submanifold that is topologically equivalent to a solid cylinder. In this case, the submanifold in question is a solid torus, which is topologically equivalent to a disk (D^2) cross a circle (S^1). So, a tubular neighborhood of this solid torus would be a neighborhood that is topologically equivalent to a solid torus (D^2 x S^1).

Now, let's consider the complement of this tubular neighborhood in S^4. This complement would be the rest of S^4 minus the neighborhood, which would include the boundary S^2 x S^1. This boundary is not topologically equivalent to a solid torus (D^2 x S^1), but rather to a solid disk (D^2) cross a circle (S^1). So, the complement is not another D^2 x S^1 x S^1, but rather a D^2 x S^1.

In conclusion, I believe that the author's statement about the complement of the tubular neighborhood being another D^2 x S^1 x S^1 is incorrect. I hope this helps clarify the situation. If you have any further questions or comments, please don't hesitate to ask. Thank you for your contribution to the forum.