Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangian submanifolds

Abstract

We establish a new version of Floer homology for monotone Lagrangian embeddings in symplectic manifolds. As applications, we get assertions for (monotone) Lagrangian submanifolds L ֒→ M which are displaceable through Hamiltonian isotopies (this happens for instance when M = C n ). We show that when L is aspherical, or more generally when the homology of its universal cover vanishes in odd degrees, its Maslov number N L equals 2. We also give topological characterisations of Lagrangians L ֒→ M with maximal Maslov number: when N L = dim(L) + 1 then L is homeomorphic to a sphere; when N L = n ≥ 6 then L fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in T * S 2k+1 whose Maslov class is zero is homeomorphic to S 2k+1 .Mathematics subject classification: 57R17, 57R58, 57R70, 53D12.

Monotone Lagrangians in of minimal Maslov number n + 1

“…Floer theory for pairs of local systems of this form is essentially equivalent to Damian's lifted Floer homology [11] on the cover L ′ . However we choose to phrase it in the above way in order to fit it into the wider context of Floer theory with local systems, closed-open string maps, and Zapolsky's orientation schemes.…”

Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces

“…The paper is organized as follows: In Section 2, we prepare fundamental properties of the Gauss images of isoparametric hypersurfaces in the standard sphere, especially the formula of their minimal Maslov number (Proposition 2.1). In Section 3, we recall Damian's lifted Floer homology and its spectral sequence [7], which is our main tool in the later sections. In Section 4, we examine the topology of Gauss images when g = 3.…”

Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

“…It is an easy application of Lemma 1 to see that for small t the entries at position (3,2) and (2, 3) are zero. Indeed, for small t the action interval of each of the two bunches narrows around the same value, but we get a lower bound from Lemma 1 (used on g , and one of the critical points) on any disc with endpoints in both-hence no such disc exists for small t. Similarly we can identify the other entries with the maps F m v i (up to homotopy) since no interaction between the two bunches at the same action level means that if we tip a little to s 1 > s 2 the two equal critical values becomes slightly different, and then for small t we get the four bunches at different action level.…”

“…3. There we basically prove that each of the bunches on the same filtration level do not interact (with respect to the differential), and that restricting the differential to any bunch is well-defined and that this always produces the same homology groups-up to a shift by the Morse index of the associated critical point of g. In fact, we will use this "bunching" construction to create a local system on N .…”

“…In fact we will identify page 1 as the Morse homology complex of g with coefficients in the local system defined in Sect. 3.…”