Exotic 4-manifolds and hyperelliptic Lefschetz fibrations

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2018

Öğrenci: TÜLİN ALTUNÖZ

Danışman: MUSTAFA KORKMAZ

Özet:

In this thesis, we explicitly construct genus-3 Lefschetz fibrations over S2 whose total space is T2 S2#6CP2 using the monodromy of Matsumoto’s genus-2 Lefschetz fibration over S2. We also present exotic minimal symplectic 4-manifolds 3CP2#kCP2 for k = 13; : : : ; 19 by twisted fiber summing of our monodromy or the genus-3 version of generalized Matsumoto’s fibration constructed by Korkmaz or by applying lantern substitutions to these twisted fiber sums. In addition, we generalize our construction of genus-3 Lefschetz fibration to genus-3k Lefschetz fibrations over S2 using the generalized Matsumoto’s genus-2k Lefschetz fibration over S2 constructed by Korkmaz and independently by Cadavid. Using the generalized version of our monodromy, we derive exotic 4-manifolds via Luttinger surgery and twisted fiber sum. Secondly, we prove that the minimal number of singular fibers in a hyperelliptic Lefschetz fibration over a sphere is 2g + 4 for even g 4 , and also, we find a lower bound for odd g 5 when the fibration is holomorphic. In addition, we discuss the number of singular fibers of a hyperelliptic Lefschetz fibration over a sphere which does not carry a complex structure.