International Congress of Mathematicians

250 P. Birannumber of steps it takes the sequence to converge to HF. In our case we have a spectralsequence starting with H*(Y'L; Z 2 ) and converging to HF(Y'L,Y'L) = 0. A computationthrough this process together with the information that Nr L > n+1 makesit possible to completely recover F*(FL;Z 2 ). It turns out that H % (TL;Z 2 ) = Z 2for i = 0,1, n and n + 1, while H 1 (YL; Z 2 ) = 0 for all 1 < i < n. Going back fromF*(FL;Z 2 ) to F*(F;Z 2 ) is now done by the Gysin exact sequence of the circlebundle Y'L —ï L and noting that the second Stiefel-Whitney class of this bundle isnothing but the restriction O\L of the generator a £ F 2 (CF"; Z 2 ).Summarizing the proof, there are three main ingredients:1. Transforming the Lagrangian F into a related Lagrangian F^ living in a differentmanifold such that F^ can be Hamiltonianly separated from itself. Consequentlywe obtain HF(Y'L,Y'L) = 0.2. Relating HF(Y'L,Y'L) to H*(Y'L) via the theory of Floer homology (e.g. aspectral sequence).3. Passing back from H*(Y L ) to H*(L).Similar ideas work in various other cases (see [8]). For example, considerCF" x CF". This manifold has Lagrangians with H\(L;Z) = 0, e.g. CF" whichcan be embedded as the "anti-diagonal" {(z,w) £ CF" x CF"|w = z}.Theorem I. Let L c CF" x CF" be a Lagrangian with H X (L;Z) = 0. ThenH*(L;Z 2 ) — F*(CF";Z 2 ), the isomorphism being of graded algebras.Another application of this circle of ideas is for Lagrangian spheres. RecentlyLagrangian spheres have attracted special attention due to their relations to interestingsymplectic automorphisms [38, 39] and to symplectic Lefschetz pencils [15].Theorem J. 1) Let M be a closed symplectic manifold with n 2 (M) = 0, anddenote by m = dime M its complex dimension. If M x CF" (where m,n > 1) hasa Lagrangian sphere then m = n + 1 (mod 2n + 2).2) Let M = CF" x CF, m + n > 3, be endowed with the split symplectic form(n + 1)