Timeline for Two Lagrangian submanifolds with clean intersections
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 5, 2021 at 15:08 | comment | added | Filip | Right, that makes sense. | |
| Mar 4, 2021 at 20:09 | comment | added | Jonny Evans | It's true that the grading on HF doesn't make sense unless L is graded... But then again, if you use the pearl model for HF, the Floer differential equals the Morse differential (there are no discs), so in some sense there is a Z grading (all pearly trajectories increase Morse index by 1). | |
| Mar 3, 2021 at 17:18 | comment | added | Filip | Incidentally, it seems to me that the isomorphism $HF(L,L) \cong H(L)$ for an exact Lagrangian cannot be stated with $\mathbb{Z}$-gradings unless $L$ is $\mathbb{Z}$-graded (which for example happens when $H^1(L,\mathbb{Z})=0$). Is this correct? | |
| Mar 3, 2021 at 17:02 | comment | added | Filip | Thanks! indeed my question doesn't make sense on its own. (I was thinking about a collection of Lagrangians $L_1,\dots, L_n$ and the Floer product in the algebra $\oplus_{i,j} HF^*(L_i, L_j)$ but that's another topic). | |
| Feb 28, 2021 at 7:39 | comment | added | Jonny Evans | If you pick gradings on L_1 and L_2 independently then you can just adjust the grading on one of them to shift the grading on HF(L_1,L_2) however you like. | |
| Feb 27, 2021 at 23:35 | comment | added | Filip | In addition, as Seidel explains in that paper, when $H^1(L_1)=H^1(L_2)=0,$ the Floer cohomology $HF^*(L_1,L_2)$ is $\mathbb{Z}$-graded, where the grading is defined up to a shift. Is there any canonical way to choose this shift, under some special conditions on the ambient and Lagrangians? Ideally, getting the isomorphism $H^*(L_1,L_2) \cong H^*(L_1 \cap L_2)$ without a shift? | |
| Nov 15, 2019 at 1:52 | comment | added | Jonny Evans | Yes, sorry: I assumed you wanted that from the title of the question, but it's definitely worth stressing! The columns of the E_2 page are the cohomology groups of the components of the intersection (shifted vertically in some way that I can't recall). | |
| Nov 14, 2019 at 14:59 | comment | added | Filip | Thanks! And just to clarify: for this spectral sequence to exist, you need $L_1$ and $L_2$ to intersect cleanly? | |
| Nov 12, 2019 at 6:44 | history | edited | Jonny Evans | CC BY-SA 4.0 |
Made answer more explicit: when are the groups isomorphic?
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| Nov 11, 2019 at 20:39 | vote | accept | Filip | ||
| Nov 11, 2019 at 18:50 | history | answered | Jonny Evans | CC BY-SA 4.0 |