Timeline for Examples of counting holomorphic curves in cylindrical reformulation of Heegaard Floer
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2023 at 19:46 | comment | added | semper-lux | @MarcoGolla Maybe I missed that: where do we require $\partial u$ to be null-homologous? I was modeling after Figure 1, which also appears to traverse one of the generating curves. The winding around the cylinders was just fanciful, in any event! If I were to connect the points using straight lines (smoothed at corners), what is the obstruction to a curve connecting $z$ to $y$? I think that's my biggest concern here: I can't see when there are obstructions to existence of curves between generators. | |
| Dec 4, 2023 at 22:49 | comment | added | Marco Golla | It looks to me like the boundary of the "curve" you drew is not null-homologous, so there can be no such curve. The way I think of these curves is as graphs over their domains, with "spikes" at the intersections of $\alpha$- and $\beta$-curves. | |
| Dec 4, 2023 at 21:03 | comment | added | semper-lux | @MarcoGolla To be clear: if they’re showing invariance and such, you’d assume they had already defined the objects well enough to draw pictures and see curves. That’s what I meant. Let me be a little more clear! Here is a stabilization of S^3 for simplicity. There are no disks starting at z in this picture. What is wrong with the “holomorphic curve” I drew in the corresponding cylindrical version? Picture | |
| Dec 4, 2023 at 20:13 | comment | added | Marco Golla | Part of the point of that paper was to prove invariance independently of the original papers (and show that invariance is easier in this context), so it's not surprising that the comparison arrives so late in the paper. As far as "how the curves look", the way I think of them is in terms of their projection onto the Riemann surface (i.e. by looking at their associated domain), but this is also how I "see" things in the original theory. | |
| Dec 4, 2023 at 19:49 | comment | added | semper-lux | @MarcoGolla Sorry for the delay! I have looked through it. I haven't fully internalized all the details, but I don't think this quite does what I was hoping for: I was looking for clarification on how the holomorphic curves should look in the cylindrical version. I suppose I could pull back the disks in OS' version using this correspondence, but that seems a bit complicated, especially considering how late in the paper it comes? | |
| Nov 28, 2023 at 0:32 | comment | added | Marco Golla | Have you looked at Section 13 in the "cylindrical reformulation" paper? The so-called "tautological correspondence" seems to be what you're looking for: you're literally counting the same objects. | |
| S Nov 27, 2023 at 19:39 | review | First questions | |||
| Nov 27, 2023 at 19:51 | |||||
| S Nov 27, 2023 at 19:39 | history | asked | semper-lux | CC BY-SA 4.0 |