Having two closed exact Lagrangian submanifolds $L_1$ and $L_2$ that intersect cleanly inside a Liouville manifold $M$ with $c_1(M)=0,$ is there (with possibly some other conditions) any relation between $$H^*(L_1 \cap L_2)$$ and $$HF^*(L_1,L_2)?$$ In particular, (when) are they necessarily isomorphic?

$\begingroup$ The intersection $L_1\cap L_2$ is zero dimensional so there is no cohomology in positive degrees. $\endgroup$– Liviu NicolaescuNov 11, 2019 at 16:05

$\begingroup$ Not necessarily. I didn't say they intersect transversely. $\endgroup$– FilipNov 11, 2019 at 17:01

1$\begingroup$ You can see work of Abouzaid arxiv.org/abs/0904.1474 for more refined statements about the Fukaya category generated by L_1,L_2. $\endgroup$– Daniel PomerleanoNov 11, 2019 at 18:58

$\begingroup$ Great, thanks ! $\endgroup$– FilipNov 11, 2019 at 20:40
1 Answer
There is a spectral sequence from the cohomology of the intersection to the Floer cohomology. This is explained (and used) in Seidel's paper "Lagrangian 2spheres can be symplectically knotted"
https://arxiv.org/abs/math/9803083
and is based on the thesis of Pozniak, available here:
http://www.math.ethz.ch/~salamon/PREPRINTS/Pozniak.pdf
(As far as I remember, Pozniak didn't mention the spectral sequence, but it's one way of understanding his work).
Edit: In particular, to answer your last question, if the Lagrangians are exact and the intersection is connected then the spectral sequence degenerates at $E_2$ and the Floer cohomology is isomorphic to the cohomology of the intersection. If there are Jholomorphic discs (e.g. potentially when the Lagrangians are nonexact) then this might not be true. The example to bear in mind is $L_1=L_2$: the intersection is connected, and we have $HF(L,L)=H(L)$ whenever L is exact, but not in general.

$\begingroup$ Thanks! And just to clarify: for this spectral sequence to exist, you need $L_1$ and $L_2$ to intersect cleanly? $\endgroup$– FilipNov 14, 2019 at 14:59

1$\begingroup$ 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). $\endgroup$ Nov 15, 2019 at 1:52

$\begingroup$ 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? $\endgroup$– FilipFeb 27, 2021 at 23:35

1$\begingroup$ 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. $\endgroup$ Feb 28, 2021 at 7:39

1$\begingroup$ 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). $\endgroup$ Mar 4, 2021 at 20:09