Timeline for $0$-surgery of slice knots and contractible manifolds
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2023 at 3:19 | comment | added | Ryan Budney | The Wikipedia page is due to me. I wrote down the definition mostly based on conversations with Danny Ruberman. If you think I should change the emphasis, just let me know. | |
| Jan 6, 2021 at 22:30 | comment | added | user160180 | Dear Professor Ruberman, slice disks corresponding to slice knots may be different. I don't understand why all contractible manifolds built in this way have a single 0-, 1-, and 2-handle except the case the slice knot is the unknot. | |
| Dec 29, 2020 at 15:58 | comment | added | Danny Ruberman | 2. The precise meaning of the term "Mazur manifold" is not universally agreed on. I think most people say this means there's a single 0/1/2 handle, some mean the specific example in Mazur's paper, and Wikipedia claims that some say that any contractible manifold is a Mazur manifold. (I've never heard this, though.) As a point of history, we should probably say Mazur-Poenaru manifolds. | |
| Dec 29, 2020 at 15:55 | comment | added | Danny Ruberman | 1. You can use any slice disk; you have to take care with the curve you use as I described. I'd guess that most of the 4-manifolds will require > 1 1-handle, and possibly some 3-handles if your disk is not ribbon. | |
| Dec 29, 2020 at 11:19 | comment | added | user160180 | I checked the algebraic details so that everything holds. I have two more questions: 1. Why we can replace the slice disk freely? 2. These contractible manifolds are all built from one 0-handle, one 1-handle and one 2-handle, i.e., all of them are Mazur? | |
| Dec 27, 2020 at 19:51 | comment | added | Danny Ruberman | There are two knots in question. The slice knot K and the other curve $\gamma$. If you do 0-framed surgery on K, and $\gamma$ normally generates the fundamental group, then you get a homology sphere no matter what the framing is on $\gamma$. As you say, the boundary of a contractible manifold must be a homology sphere. | |
| Dec 27, 2020 at 11:22 | comment | added | user160180 | Attaching $2$-handle corresponds to doing integral surgery. Let $Y$ be the $3$-manifold which is obtained by $0$-surgery on a slice knot. We also need to assume that the resulting $3$-manifold which is obtained by surgery on a knot in $Y$ is a homology sphere, right? Or it is a consequence of having contractible $4$-manifold? | |
| Dec 26, 2020 at 22:12 | vote | accept | CommunityBot | ||
| Dec 26, 2020 at 21:49 | comment | added | Danny Ruberman | Yes, providing that the condition I mentioned is satisfied: the homotopy class of $\gamma$ must normally generate the fundamental group of the 0-surgered manifold. | |
| Dec 26, 2020 at 21:35 | comment | added | user160180 | Thus, if $\gamma$ is a knot in the $3$-manifold obtained by $0$-surgery on a slice knot, and if we attach $2$-handle along $\gamma$, we produce a contractible manifold, right? | |
| Dec 25, 2020 at 14:10 | history | edited | Danny Ruberman | CC BY-SA 4.0 |
added 31 characters in body
|
| Dec 24, 2020 at 23:29 | history | answered | Danny Ruberman | CC BY-SA 4.0 |