Symplectic isotopy conjecture for elliptic ruled surfaces.

Three problems are studied in this thesis; the first problem is about four-dimensional symplectic manifolds. It was formulated by McDuff and Salamon in the very latest edition of their famous book. This problem is to prove that the Torelli part of the symplectic mapping class group of a geometrically ruled surface is trivial. In Section 0 a partial solution for this problem is given. The second problem is to compute the symplectic mapping class group of the one-point blow-up of the direct product of the 2-sphere and the 2-torus. A partial solution to this problem is given in Section 3, see also my joint work with Shevchishin. Namely, it is proved that the abelianization of the corresponding symplectic mapping class group is of rank 2. The third problem has nothing to do with symplectic geometry, it is purely topological. This problem studies necessary and sufficient conditions for the existence of Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the cobordism is the spin Lorentzian group. This extends a result of Gibbons-Hawking on Sl(2, C)-Lorentzian cobordisms between 3-manifolds and results of Reinhart and Sorkin on the existence of Lorentzian cobordisms. The proof is explained very carefully in my recent joint work with Rafael Torres. Here the explanation tends to be briefly.