Noncomplex Smooth 4-manifolds with Lefschetz Fibrations

Recently, B. Ozbagci and A. Stipsicz [12] proved that there are infinitely many pairwise nonhomeomorphic 4-manifolds admitting genus-2 Lefschetz fibration over S but not carrying any complex structure with either orientation. (For the definition of Lefschetz fibration, see [6].) Their result depends on a relation in the mapping class group of a closed orientable surface of genus 2. This relation with eight right Dehn twists was discovered by Y. Matsumoto [9] by a computer calculation, and it is the global monodromy of a Lefschetz fibration T ×S#4CP 2 → S, where S is the 2-sphere and T 2 is the 2-torus. In this paper, we generalize Matsumoto’s relation to higher genus orientable surfaces. We find a relation involving 2g + 4 (resp., 2g + 10) Dehn twists when the genus of the surface is even (resp., odd). Following the method of Ozbagci and Stipsicz, for every positive integer n, we obtain a 4-manifold Xn admitting a genus-g Lefschetz fibration such that the fundamental group of Xn is isomorphic to Z ⊕ Zn for every g ≥ 2. We then deduce that the 4-manifold Xn does not admit any complex structure. This is the main result of this paper.