Timeline for Spin structure on mapping torus
Current License: CC BY-SA 3.0
11 events
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| Jun 14, 2018 at 16:06 | comment | added | Michael Albanese | @OscarRandal-Williams: Thanks. So this is a specifically two-dimensional phenomena. I was hoping to find a similar action of $SL(3, \mathbb{Z})$ on the spin structures on $T^3$. | |
| Jun 14, 2018 at 15:15 | comment | added | Oscar Randal-Williams | @MichaelAlbanese: I think I worked this out when reading "Spin structures and quadratic forms on surfaces" by Dennis Johnson, which shows that Spin structures are the same as quadratic refinements of the intersection form; the mapping class group acts on quadratic refinements by precomposition, and I think it is then easy to derive the above formula. | |
| Jun 14, 2018 at 13:27 | comment | added | Michael Albanese | @OscarRandal-Williams: Can you give a reference for this action or give an indication of how to derive it? | |
| May 8, 2014 at 21:30 | comment | added | Jan Jitse Venselaar | Ah, that makes sense, the identification of the class of spin structures with $\mathbb{Z}/2 \oplus \mathbb{Z}/2$ was different from the one I had in my head: here (1,1) corresponds to the trivial one, I was using (0,0) for the trivial one. | |
| May 8, 2014 at 21:08 | comment | added | Oscar Randal-Williams | @Jan: I think that $(u,v)=(1,1)$ is fixed by any operation (the Arf invariant I have in mind is $(1+u)(1+v)$). This uses that if $A,B,C,D$ are the entries of an element of $SL(2,\mathbb{Z})$ then $A+C+AC \equiv 1 \, mod \, 2$ (i.e. $A$ and $C$ cannot both be even). | |
| May 8, 2014 at 19:51 | comment | added | Jan Jitse Venselaar | @OscarRandal-Williams : do you have a reference for that formula? It seems to be at odds with the transformation rules from projecteuclid.org/euclid.cmp/1104115859, in the sense that your formula allows you to go from any spin structure to any other spin structure, while there it is shown that "even" and "odd" spin structures are distinct under the action of diffeomorphisms (see around 4.25 in the paper), and there is 1 odd spin structure on the torus (the trivial one), and the other 3 are even. | |
| May 8, 2014 at 17:06 | comment | added | Xiao-Gang Wen | @ Kevin Walker @ Oscar Randal-Williams : I would like to ask if all 4-dimensional compact orientable mapping tori are spin? | |
| Aug 15, 2011 at 7:04 | vote | accept | Dylan | ||
| Aug 15, 2011 at 7:04 | |||||
| Aug 12, 2011 at 22:46 | comment | added | Oscar Randal-Williams | Ah, I had missed "naturally". In that case: what Kevin said. | |
| Aug 12, 2011 at 16:42 | comment | added | Kevin Walker | But even if $f$ preserves the spin structure, there are two spin structures on the mapping torus which restrict to the original spin structure on the fiber. So there's no way to "naturally endow" the mapping torus with a spin structure (as in the original question). Unless of course you choose a lifting of $f$ to a map of spin bundles. | |
| Aug 12, 2011 at 13:49 | history | answered | Oscar Randal-Williams | CC BY-SA 3.0 |