Timeline for Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 28, 2020 at 4:19 | comment | added | Joe | It's actually showed $RP^4 \# CP^2$ and $Q \# CP^2$ ($Q$ being the fake $RP^4$) are diffeomorphic in this article: [S. Akbulut, A fake 4-manifold, 4-Manifold Topology, Contemporary Math. 35 (1984), 75-141.] | |
| Jun 18, 2020 at 15:53 | comment | added | Arun Debray | For these specific $M$ and $N$, I think $X := \mathbb{CP}^2 \# (S^2\times S^2)$ works: for some $i$ and $j$, $M \#_i X$ and $M \#_j X$ are diffeomorphic. This is because $M \# \mathbb{CP}^2$ and $N \#\mathbb{CP}^2$ are $S^2\times S^2$-stably diffeomorphic: one once again uses Kreck's modified surgery theory, but $M \# \mathbb{CP}^2$ and $N \# \mathbb{CP}^2$ aren't pin+ or pin-, so the calculation takes place in the 4th unoriented bordism group, where they are bordant. | |
| Jun 18, 2020 at 15:47 | comment | added | Arun Debray | That's a great question, and I don't know what happens in general. The trick that makes this work is that $S^2\times S^2$ is null-bordant, allowing bordism to be used to check stable diffeomorphism. So something different would have to be done for $\mathbb{CP}^2$. | |
| Jun 18, 2020 at 15:20 | comment | added | Bruno Martelli | Aha, I didn't know that. What happens if we replace $S^2 \times S^2$ with some other fixed 4-manifold $X$, like for instance the complex projective plane? Are the connected sums $M\#X$ and $N\# X$ always guaranteed to be non-diffeomorphic for these two specific 4-manifolds $M$ and $N$? | |
| Jun 17, 2020 at 15:47 | comment | added | Arun Debray | It is often but not always true that homeomorphic $M$ and $N$ are $S^2\times S^2$-stably diffeomorphic. (It is true when $M$ and $N$ are oriented, by a theorem of Gompf.) One counterexample is Capell-Shaneson's fake $\mathbb{RP}^4$ and the standard $\mathbb{RP}^4$, in fact, which can be shown with Kreck's modified surgery theory, reducing the problem to the possible values they can take on in the 4th pin+ bordism group. | |
| Jun 17, 2020 at 14:29 | history | answered | Bruno Martelli | CC BY-SA 4.0 |