Timeline for Building examples of elements of $\Omega_4(\xi)$ via surgery theory: how to do it?
Current License: CC BY-SA 3.0
17 events
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| Jan 13, 2017 at 13:12 | comment | added | Riccardo | @user95545 thanks a lot for this clarification, it's enlightening | |
| Jan 13, 2017 at 13:07 | comment | added | user95545 | Surgery over $\xi$ does not change the bordism class, so cannot produce the desired bordism class unless you had it already. For the determination of the bordism group I would suggest you try an Adams spectral sequence here. | |
| Jan 13, 2017 at 13:06 | comment | added | user95545 | The more standard way of using surgery would be to first decide on a bordism class in $\Omega_4(\xi)$ and then to use surgery to find a particularly nice representative in the bordism class. "It's an element of $\Omega_4(\xi)$" is not the same as "has normal 1-type $\xi$". Here you might be able to use surgery if I understand @RyanBudney "dihedral groups are good for surgery theory in dimension 4" correctly. | |
| Jan 13, 2017 at 12:21 | comment | added | Riccardo | and for what concern your suggestion, I think I need a little bit of time to digest is due to my "ignorance" about surgery. So if I'm understanding your suggestion (please correct me), you suggest to create a $2$-equivalence $(f,\nu_N) \colon N \to K(D_{2n},1)\times BSO$ (or $BSpin$?) and then $N$ should satisfy the properties I'm looking for? (maybe my last sentence makes little sense, but I'm trying to understand even how surgery might help me) | |
| Jan 13, 2017 at 12:15 | history | edited | Riccardo | CC BY-SA 3.0 |
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| Jan 13, 2017 at 12:14 | comment | added | Riccardo | @MarkGrant oh I think that in the way of editing my question to make it clearer I omitted the part where $n= 0 \pmod{4}$, which is the hardest (and most interesting part). All the computations I mentioned here are done in this case (I was able to solve the other cases ) | |
| Jan 13, 2017 at 11:23 | comment | added | Mark Grant | (I'm a bit nervous about the low dimensions here). Does this help in any way? | |
| Jan 13, 2017 at 11:22 | comment | added | Mark Grant | A few comments: When $n$ is odd, it seems that $H_4(D_{2n})=0$ (quoting this page: groupprops.subwiki.org/wiki/Group_cohomology_of_dihedral_groups) so what you ask for can't occur. When $n$ is even, take a non-trivial element of $H_4(D_{2n})\cong \mathbb{Z}/2\oplus\mathbb{Z}/2$ and ask whether it is realized by a $4$-manifold with the properties you want. I think it will be realized by a map $f:N\to K(D_{2n},1)$ from an orientable manifold $N$. Now, perhaps you can perform surgery on the map $(f,\nu_N):N\to K(D_{2n},1)\times BSO$ to make it a $2$-equivalence... | |
| Jan 13, 2017 at 10:33 | history | edited | Riccardo | CC BY-SA 3.0 |
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| Jan 12, 2017 at 10:49 | comment | added | Riccardo | Dear @RyanBudney, sorry for writing you again. I gave a more deep read to the paper you linked and I still wasn't able to find an application to surgery which could help me. If i understood correctly, lots of the results listed there required properties on the fundamental group that I don't think I have, PD_r, virtually free, torsion-free... and in chapter 6.3 (where they speak about stable classification), they just mention some general facts but they don't make use of surgery. Could you please point me out where should I focus in such a paper? thanks in advance! | |
| Jan 9, 2017 at 9:46 | history | edited | Riccardo | CC BY-SA 3.0 |
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| Jan 8, 2017 at 22:06 | history | edited | Riccardo | CC BY-SA 3.0 |
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| Jan 8, 2017 at 9:55 | comment | added | Riccardo | @RyanBudney I'm happy you gave me a reference rather than a direct answer! I skimmed through the index and notice that it contains a lot of material. Could you please point me out the chapters I should focus? I will start to read it now from the beginning, but since I am a kind of "novice" in this field I don't want to miss the results I need only because they are expressed in a more involved or different way. Again, thanks a lot for the link | |
| Jan 8, 2017 at 8:44 | comment | added | Ryan Budney | Dihedral groups are good for surgery theory in dimension 4. Rather than give you a reference to the results you need, here is a reference to a paper that uses them (roughly) in the way you want: maths.usyd.edu.au/u/jonh/fmgk16.pdf | |
| Jan 8, 2017 at 8:18 | history | edited | Riccardo |
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| Jan 7, 2017 at 21:57 | history | edited | Riccardo | CC BY-SA 3.0 |
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| Jan 7, 2017 at 18:07 | history | asked | Riccardo | CC BY-SA 3.0 |