International Congress of Mathematicians

248 P. Birangroup of Hamiltonian diffeomorphisms has infinite diameter with respect to Hofer'smetric. Note that when 7Ti(Ham(M,u;)) is finite the same holds also for Ham(M,w)itself. (See [32] for the details and references for other results on the diameter ofHam). This is applicable when (M,OJ) contains a Lagrangian submanifold A withHF(A,A) # 0, since then HF(0 S i x A, 0 S i x A) = (Z 2 ®Z 2 ) ® H F(A, A) # 0. Forexample, taking A = RF" c CF" Polterovich proved that diamHam(CF") = oo(for n= 1,2 the same holds for diaroHam(CF")).In view of the above the following question seems natural: does every closedsymplectic manifold contain a subset A with open non-empty complement and withthe stable Hamiltonian intersection property ? Note that besides Lagrangian submanifolds(with H F ^ 0) no other stable Hamiltonian intersection phenomena areknown. It would also be interesting to find out whether the intersections describedin Theorems C,D,E and especially E' continue to hold after stabilization.4. Intersections versus non-intersectionsIn contrast to cotangent bundles there are manifolds in which every compactsubset can be separated from itself by a Hamiltonian isotopy. The simplest exampleis C" : indeed linear translations are Hamiltonian, and any compact subset can betranslated away from itself. Clearly the same also holds for every symplectic manifoldof the type M x C by applying translations on the C factor. Note that manifoldsof the type M x C sometime appear in "disguised" forms (e.g. as subcriticai Steinmanifolds, see Cieliebak [14]).The "non-intersections" property has quite strong consequences on the topologyof Lagrangian submanifolds already in C". Denote by u; s td the standard symplecticstructure of C" and let A be any primitive of uj s td- Note that the restrictionA|T(L) °f A to any Lagrangian submanifold F c C" is closed. The following wasproved by Gromov in [22]:Theorem G. Let F c C" be a Lagrangian submanifold. Then the restriction of Xto L is not exact. In particular H 1^; R) ^ 0.Indeed if A were exact on F then A u : iT2(C n ,L) —t R must vanish, hence by-Theorem F it is impossible to separate F from itself by a Hamiltonian isotopy. Onthe other hand, as discussed above, in C" this is always possible. We thus get acontradiction. (Gromov's original proof is somewhat different, however a carefulinspection shows it uses the failure of Lagrangian intersections in an indirect way).Arguments exploiting non-intersections were further used in clever ways by Lalondeand Sikorav [25] to obtain information on the topology of exact Lagrangians incotangent bundles (see also Viterbo [42] for further results).An important property of symplectic manifolds W having the "non-intersections"property is the following vanishing principle: for every Lagrangian submanifoldL c W with well defined Floer homology we have HF(L, L) = 0. Applying thisvanishing to C" yields restrictions on the possible Maslov class of Lagrangian submanifoldsof C". (Conjectures about the Maslov class due to Audin appear alreadyin [1]. First results in this directions are due to Polterovich [31] and to Viterbo [41].The interpretation in Floer-homological terms is due to Oh [29]. Generalizations