Timeline for Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles
Current License: CC BY-SA 3.0
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| Sep 15, 2023 at 12:05 | comment | added | Danny Ruberman | @MohithRaju The double covering $SU(2) \to SO(3)$ exhibits $SU(2)$ as the spin group $Spin(3)$. So what I said is a special case of the general principle that $w_2$ is the obstruction to existence of a spin structure. You can read about this many places, starting with Wikipedia en.wikipedia.org/wiki/Spin_structure. More comprehensive treatments are in books like Lawson-Michelson Spin Geometry or Friedrichs Dirac Operators in Riemannian Geometry or lots of informal lecture notes. Try googling "spin structure". | |
| Sep 14, 2023 at 5:00 | comment | added | Mohith Nagaraju | Could you please expand/ provide a reference for the statement: "In particular, $SO(3)$ bundles may have non-trivial $w_2$, which obstructs their lifting to an $SU(2)$ bundle." | |
| Sep 12, 2018 at 16:55 | comment | added | Danny Ruberman | I was referring to avoiding non-compactness due to `bubbling'. So for instance in the Fintushel-Stern paper, they look at self-dual connections on a bundle with $p_1 \in [0,4]$ (and non-trivial $w_2$ if $p_1 =4$). Something similar happens in Morgan-Mrowka, A note on Donaldson's polynomial invariants, IMRN 1992. See also the proof of Floer's "excision theorem" in Braam-Donaldson, The Floer memorial volume, 195–256 where the lack of reducibles for SO(3) connections on a torus is crucial. | |
| Sep 12, 2018 at 10:39 | comment | added | Gorbz | what are the "many circumstances"? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 12, 2015 at 13:55 | vote | accept | Bilateral | ||
| Mar 12, 2015 at 11:30 | history | answered | Danny Ruberman | CC BY-SA 3.0 |