Resumen de Relative family Gromov¿Witten invariants and symplectomorphisms

Olguta Buse

• We study the symplectomorphism groups G? = Symp0(M,??) of a closed manifold M equipped with a one-parameter family of symplectic forms ?? with variable cohomology class. We show that the existence of nontrivial elements in p*(A,A'), where (A,A') is a suitable pair of spaces of almost complex structures, implies the existence of nontrivial elements in p*-i(G?), for i = 1 or 2. Suitable parametric Gromov¿Witten invariants detect nontrivial elements in p*(A,A'). By looking at certain resolutions of quotient singularities we investigate the situation (M,??) = (S2 × S2 × X,sF ? ?sB ? ?arb), with (X,?arb) an arbitrary symplectic manifold. We ?nd nontrivial elements in higher homotopy groups of G?X, for various values of ?. In particular we show that the fragile elements wl found by Abreu and McDuff in p4l(Gl+1pt) do not disappear when we consider them in S2 × S2 × X.