Timeline for On the smoothness of plumbed $4$-manifolds and $E_8$ manifold
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 17, 2020 at 3:08 | comment | added | Ben Wieland | Before Donaldson, there was Rokhlin's theorem, that the signature of the intersection form of a smooth closed spin 4-manifold is divisible by 16 (rather than just 8 as the arithmetic implies). This is enough to show that the $E_8$ manifold is not smoothable. | |
| Nov 17, 2020 at 2:38 | comment | added | LSpice | Your link "Freedman" originally pointed to Wikipedia. I assumed the intended target was his article referenced there, so I edited the link (and added the title of the article). | |
| Nov 17, 2020 at 2:37 | history | edited | LSpice | CC BY-SA 4.0 |
Wikipedia link to paper link; other minor link edits
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| Nov 16, 2020 at 21:16 | comment | added | user160180 | And the plumbing $4$-manifold with $E_8$ and Freedman's $E_8$ manifold are not same. | |
| Nov 16, 2020 at 21:12 | comment | added | user160180 | So, the critical word is "closed", right? | |
| Nov 16, 2020 at 21:04 | comment | added | Rohil Prasad | Donaldson's diagonalization theorem implies any closed smooth four-manifold cannot have intersection form $E_8$. The plumbing only gives a smooth four-manifold with boundary with intersection form $E_8$. However, Freedman showed that any integral homology three-sphere bounds a compact contractible topological four-manifold, so you can cap off the Poincare sphere boundary to get a topological four-manifold with intersection form $E_8$, which is usually what people call the "$E_8$-manifold". | |
| Nov 16, 2020 at 21:01 | comment | added | user160180 | Poincare homology sphere should be the boundary of $E_8$. I don't follow the part "Freedman's construction". Could you explain it more? | |
| Nov 16, 2020 at 20:54 | comment | added | Rohil Prasad | The plumbing along the $E_8$ graph yields a smooth four-manifold with intersection form $E_8$ that bounds the Poincare homology sphere. Freedman constructed the $E_8$ manifold from this by attaching a compact contractible topological four-manifold that bounds the Poincare homology sphere. Donaldson's theorem essentially implies you can't do this smoothly. | |
| Nov 16, 2020 at 20:45 | history | asked | user160180 | CC BY-SA 4.0 |