Speaker:  Adalet Çengel (Boğaziçi University)

Title:  Signatures of Lefschetz fibrations

Abstract: Donaldson has shown that a closed symplectic $4$-manifold up to blow-up is equivalent to that of a Lefschetz fibration over the $2$-sphere. The topology of the total space of a Lefschetz fibration is completely determined by its monodromy representation which is a product of positive Dehn twists.
We give an algorithm to compute signature of a given Lefschetz fibration over $2$-disk by using its monodromy factorization. Our main tool will be Wall’s non-additivity formula applied to what we call partial fiber sum decomposition of a Lefschetz fibration over disk. We show that our algorithm works for Lefschetz fibrations with regular fiber having nonempty boundary. When the regular fibers are closed, it is a reformulation of Burak Özbağcı’s algorithm which is described in his Ph.D. thesis for the calculation of signatures of Lefschetz fibrations. This is a joint work with Çağrı Karakurt.