Timeline for Homology spheres bounding homology balls but not embedding into $S^4$
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Oct 31, 2022 at 3:08 | answer | added | mme | timeline score: 4 | |
| Nov 15, 2018 at 23:10 | comment | added | David Snyder | @user101010 Right, Kervaire's example is simply connected, definitely not contractible (has non-trivial $H_2$), I had to go back to the paper (and in hindsight that it was not contractible should have been obvious). Still, the Note that Freedman makes immediately after stating his own Theorem 2 (homology 3-spheres embed top. flatly in S^4) does state explicitly that the topological ball bounded by the Poincaré sphere can't be smoothed, as shown by the Rochlin invariant --this was the observation I was writing about, not the much harder result (Theorem 2), indeed far from being an observation. | |
| Nov 15, 2018 at 19:35 | comment | added | user101010 | @DavidSnyder I didn’t check the paper but my guess is that kervaire showed that the poincare homology sphere can’t bound any smooth homology ball (by Rochlin’s theorem) but certainly didn’t construct any topological one. All that stuff is freedman - and it appears to be pretty far from “observed” caliber. | |
| Nov 15, 2018 at 17:07 | comment | added | David Snyder | So, Kervaire's example isn't smoothable (though topologically flat) which was basically observed by Freedman in his paper. | |
| Nov 15, 2018 at 16:51 | comment | added | David Snyder | @MikeMiller Ah, right! thanks, that's not my usual world. | |
| Nov 14, 2018 at 21:33 | comment | added | mme | @DavidSnyder He is talking about bounding a smooth manifold. The Freedman result as cited is true. | |
| Nov 14, 2018 at 17:30 | comment | added | David Snyder | @user1010 Not quite. Kervaire (1969) gives an example of an integral homology $3$-sphere that does not bound a contractible $4$-manifold. (see page 69) | |
| Nov 14, 2018 at 5:16 | comment | added | user101010 | @DavidSnyder I believe every 3-dimensional homology sphere bounds a topological contractible 4-manifold (But this is a hard theorem of Freedman). By doubling this we get a 4-sphere (again - a theorem of freedman) | |
| Nov 13, 2018 at 23:17 | comment | added | David Snyder | The Poincaré homology $3$-sphere does not smoothly embed in the $4$-sphere but does have a topologically bi-collared embedding. Hence, it bounds topological (but not smooth) homology $4$-balls to either side. | |
| Nov 13, 2018 at 3:40 | comment | added | Ryan Budney | 3-dimensional punctured homology spheres embedding in $S^4$. There are examples, of course, but we don't know which ones embed and which ones do not. Similarly there are examples of un-punctured homology spheres embedding, but we don't know which ones embed, and which ones do not. | |
| Nov 12, 2018 at 21:40 | comment | added | user101010 | @RyanBudney Thanks for the follow up. I am confused, are you talking about a 4-dimensional or a 3-dimensional punctured homology sphere embedding in $S^4$? Aren't there lots of examples of 3-dimensional homology spheres embedding in $S^4$ - so any of those you could just puncture. And in the 4-dimensional case - wouldn't this be false assuming Schoenflies? | |
| Nov 9, 2018 at 14:05 | comment | added | mme | @Ryan Right, by passing to the boundary of a tubular neighborhood that is equivalent to the question of whether $\Sigma \# \overline{\Sigma}$ embeds. | |
| Nov 9, 2018 at 8:48 | comment | added | Ryan Budney | I talked with Livingston and he doesn't think he and Gilmer deserve credit for the question. He suggested the weaker question is also interesting: whether or not a punctured homology sphere smoothly embeds in S^4. | |
| Nov 7, 2018 at 16:40 | comment | added | mme | @Ryan Their paper ("On embedding 3-manifolds...) was one of the first places I checked, but no dice. They certainly did the Mayer-Vietoris argument to check that the embedding implied bounding homology balls (in their case, rational). I wouldn't be surprised if they vocalized this thought at some point. Also, Kirby Problem 3.20 is to characterize which 3-manifolds embed in 4-space; it observes that there are obstructions to bounding homology balls, but doesn't "cut the problem in half" like we want. | |
| Nov 7, 2018 at 3:47 | comment | added | Ryan Budney | I talked with Danny a couple hours ago. He seems to think the issue came up in Gilmer and Livingston's work in the 80's. | |
| Nov 7, 2018 at 0:00 | comment | added | mme | @RyanBudney That's a good question. I looked for a while but had no luck; I'll ask around (or maybe Danny Ruberman will see this question and know). If it helps anyone, my standard reference for existence results and discussion of this problem is your and Burton's "Census..." As a side comment, one natural place to try to look for counterexamples is $\Sigma \# \overline \Sigma$. But it is a theorem of Gompf, nicely written up in Kyle Larson's thesis, that if $\Sigma$ is $\pm 1/n$-surgery on a knot, that manifold does embed. | |
| Nov 6, 2018 at 23:15 | comment | added | Ryan Budney | @MikeMiller: do you know who might have first formulated that problem, and where? If I were to guess I'd say perhaps Fintushel and Stern, perhaps a long ways back, but I'm not certain which paper to look in. | |
| Nov 6, 2018 at 23:08 | comment | added | user101010 | @Mike Miller Thanks Mike - that is what I was imagining. | |
| Nov 6, 2018 at 22:50 | comment | added | mme | This is a major open problem. | |
| Nov 6, 2018 at 22:45 | history | asked | user101010 | CC BY-SA 4.0 |