Timeline for Combinatorial spin$^{\mathbf{C}}$ structures
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 8, 2017 at 4:29 | comment | added | Zitao Wang | Thanks for the reference, Ryan. The combinatorial representation in v2 of the paper looks somewhat obscure to me. For the spin case, it should be equivalent to the Kasteleyn orientation I mentioned above, though not in a way that's clear to me. It would be great if this connection is point out explicitly in the new version and that the combinatorial representation for spin$^{\mathbf{C}}$ can also be written in a way similar to the Kasteleyn orientation. | |
| Nov 8, 2017 at 3:45 | comment | added | Ryan Budney | The revised version (v3) should appear on the arXiv tomorrow. Hopefully it's far less painful to read. | |
| Nov 8, 2017 at 3:35 | comment | added | Ryan Budney | Yes, there are similar combinatorial representations. The 2nd link you give cites my preprint. The last section of my preprint shows how you adapt the first part of the paper to describe $Spin^c$ structures in the same language. I'll update my preprint soon as I think (sigh!) the version on the arXiv is still difficult to read in spots. | |
| Nov 8, 2017 at 2:00 | comment | added | HYL | In general $\text{spin}^{\mathbf{C}}$-structures correspond to homotopy classes of complex structures over the 2-skeleton which can be extended to the 3-skeleton. So for surfaces, $\text{spin}^{\mathbf{C}}$-structures correspond to homotopy classes of complex structures on the triangulation. | |
| Nov 7, 2017 at 20:00 | comment | added | Zitao Wang | 2D means two dimension. Yes, the spin$^{\mathbf{C}}$ structures are integral lifts of $w_2$. But I'm not sure how to translate this into the orientations of the edges of the graph on the manifold. i.e., something analogous to the Kasteleyn orientation. | |
| Nov 7, 2017 at 19:51 | history | edited | Zitao Wang | CC BY-SA 3.0 |
added 24 characters in body
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| Nov 7, 2017 at 19:26 | comment | added | Alex Degtyarev | Is $2D$ two or even? As far as I remember, $\operatorname{Spin}^{\mathbb{C}}$-structures are the integral lifts of $w_2$. If $2D=2$ and, hence, $w_2=0$, there is a distinguished lift, do the set is naturally isomorphic to $\mathbb{Z}$. | |
| Nov 7, 2017 at 19:12 | history | asked | Zitao Wang | CC BY-SA 3.0 |