MATHEMATICS SEMINAR by Tülin Altunöz Author:COLLEGE OF SCIENCESTime: 10:00Location: SCI 103
For the past two decades the category of 4-dimensional manifold has experienced ex- plosive growth. A number of surprising results, which exhibits rich and complicated relationships between di_erent categories of manifold which are unique to dimension 4, has been discovered associated with this growth. These developments have made symplectic 4-manifolds that are another category of 4-dimensional manifolds natu- ral candidates to be the building blocks of all smooth 4-manifolds having in_nitely di_erentiable local identi_cation maps.
Donaldson and Gompf results ([1], [2], [3] and [4]) give the relation between sym- plectic 4-manifolds and Lefschetz _brations, which are a _bering of a 4-manifold by surfaces, with a _nite number of singularities of a prescribed type. Their results say that symplectic 4-manifolds (after perhaps blowing up) admit the structure of a Lefschetz _bration and a genus-g Lefschetz _bration with a _ber genus g _ 2 over the Riemann surface admits a symplectic structure. Hence, Lefschetz _brations provide a combinatorial way to study symplectic 4-manifolds. The results on the minimal number of vanishing cycles of a Lefschetz _bration provide some results on symplectic 4- manifolds, which also gives a connection between symplectic topology and geometric group theory.
The information about the number of singular _bers in a Lefschetz _bration provides us important information about the topological invariants of its total space such as _(X), e(X), c21 (X) and so on. In addition, it has been known that the number of singular _bers in a Lefschetz _bration can not be arbitrary. So it makes sense to ask what the minimal number of singular _bers in a nontrivial relatively minimal genus-g Lefschetz _bration over the oriented surface of genus h.
In this talk, after giving some results about the minimal number of singular _bers in a nontrivial relatively minimal genus-g Lefschetz _bration over the surface of genus h, we will state our results about it for hyperelliptic Lefschetz _brations which are a special kind of Lefschetz _brations.
References [1] S.K.Donaldson, \ Lefschetz _brations is symplectic geometry", In Proceedings of the International Congress of Mathematicians (Berlin,1998), Vol.II, Doc. Math., Extra Vol. ICM Berlin, 1998, 309-314. 272. [2] S.K.Donaldson, \Lefschetz pencils on symplectic manifolds", J. Di_erential Geom. 53 (1999), 205-236. 272. [3] R.E. Gompf,\The topology of symplectic manifolds", Turkish J. Math. 25 (2001), 43-59. 272,278. [4] R.E. Gompf and A.I.Stipsicz,\4-manifolds and Kirby calculus",Grad. Stud. Math. 20, Amer. Math. Soc., Providence; R.I., 1999. 272, 278, 290. |