Time: 10:00
Location: SCI 103


Speaker          : Tülin Altunöz
Title                : The Number of Singular Fibres in Hyperelliptic Lefschetz Fibrations

Date                : January 17, 2019, Thursday
Time               : 10:00
Cookie&Tea  : SCI 103 09:45
Place               : SCI 103

Abstract         :  

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.



[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.