International Congress of Mathematicians

Geometry of Symplectic Intersections 249to other manifolds appear in [2] and [11]. Finally, consult [21] for recent resultsanswering old questions on the Maslov class).4.1. Lagrangian embeddings in closed manifoldsThe ideas described above can be applied to obtain information on the topologyof Lagrangian submanifolds of some closed manifolds. Note that in comparisonto closed manifolds the case of C" can be regarded as local (Darboux Theorem). Ofcourse, "local" should by no means be interpreted as easy. On the contrary, characterizationof manifolds that admit Lagrangian embeddings into C" is completelyout of reach with the currently available tools.Below we shall deal with the "global" case, namely with Lagrangians in closedmanifolds. One (coarse) way to "mod out" local Lagrangians is to restrict to LagrangiansF with H\(L;Z) zero or torsion (so that by Theorem G they cannot liein a Darboux chart). The pattern arising in the theorems below is that under suchassumptions in some closed symplectic manifolds we have homological uniquenessof Lagrangian submanifolds. Let us view some examples.We start with CF". It is known that a Lagrangian submanifold F c CF"cannot have H\(L;Z) = 0 (see Seidel [39], see also [10] for an alternative proof).However, F c CF" may have torsion H\(L;Z) as the example RF" c CF" shows.Theorem H. Let L c CF" be a Lagrangian submanifold with H\(L; Z) a 2-torsiongroup (namely, 2H\(L;Z) = 0). Then:1. H*(L;Z 2 ) = F*(RF";Z 2 ) as graded vector spaces.2. Let a £ F 2 (CF";Z 2 ) be the generator. Then O\L £ H 2 (L;Z 2 ) generates thesubalgebra F even (F;Z 2 ). Moreover if n is even the isomorphism in 1 is ofgraded algebras.Statement 1 of the theorem was first proved by Seidel [39]. An alternativeproof based on "non-intersections" can be found in [8]. Let us outline the mainideas from [8]. Consider CF" as a hypersurface of CF" +1 . Let U be a smalltubular neighbourhood of CF" inside CF" +1 . The boundary dU looks like a circlebundle over CF" (in this case it is just the Hopf fibration). Denote by F^ —t Lthe restriction of this circle bundle to F c CF". A local computation shows thatU can be chosen so that F^ c CF" +1 \ CF" becomes a Lagrangian submanifold.(This procedure works whenever we have a symplectic manifold S embedded as ahyperplane section in some other symplectic manifolds M). The next observationis that Y'L C CF" +1 \ CF" is monotone and moreover its minimal Maslov numberNr L is the same as the one of F. Due to our assumptions on H\(L;Z) this numberturns out to satisfy Nr L > n + 1. The crucial point now is that HF(Y'L,Y'L) = 0.Indeed, the symplectic manifold CF" +1 \ CF" can be completed to be C" +1 whereFloer homology vanishes.Having this vanishing we turn to an alternative computation of HF(Y'L,Y'L).This computation is based on the theory developed by Oh [29] for monotone Lagrangiansubmanifolds. According to [29] Floer homology can be computed via aspectral sequence whose first stage is the singular cohomology of the Lagrangian.The minimal Maslov number has an influence both on the grading as well as on the