Small Lefschetz fibrations and exotic 4-manifolds

Abstract

Abstract. We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP 2 #p CP 2 for p = 7, 8, 9, and to 3CP 2 #q CP 2 for q = 12, . . . , 19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b + ≤ 3, with one possible exception when b + = 3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smallest possible example, which follow from a reverse engineering procedure we introduce for this setting. We also derive exotic minimal symplectic 4-manifolds in the homeomorphism classes of CP 2 #4CP 2 and 3CP 2 #6CP 2 from small Lefschetz fibrations over surfaces of higher genera.

Minimal number of singular fibers in Lefschetz fibrations over the torus

“…By modifying the 4-chain relation, Baykur and Korkmaz [1] found an interesting relation in M 1 2 . In the following we lift the Baykur-Korkmaz relation (a relation which is obtained from Equation (3.1) below by capping off the boundary circle δ 0 with a disk) to M 2 2 , and using this lift we improve the upper bounds for N (3,1) and N (4, 1).…”

“…In the following we lift the Baykur-Korkmaz relation (a relation which is obtained from Equation (3.1) below by capping off the boundary circle δ 0 with a disk) to M 2 2 , and using this lift we improve the upper bounds for N (3,1) and N (4, 1). This result was predicted by Hamada and a sketch is given in [12].…”

“…Recently Baykur and Korkmaz [1] found an interesting relation in the mapping class group M 1 2 of the genus-2 surface with one boundary component, and by using this relation they showed that N (2, 0) = 7. Furthermore, using the 8-holed torus relation [15] and the Matsumoto-Cadavid-Korkmaz relation [3,13,19], Hamada [12] found an upper bound for N (g, 1): he showed that N (g, 1) ≤ 4 if g ≥ 5 and N (g, 1) ≤ 6 if g = 3, 4.…”

Fillings of Genus–1 Open Books and 4–Braids

“…Two ingredients in this simple construction are suitable partial conjugations which effectively change the homology (e.g. [29,6,5]), and the explicit monodromy of a genus-1 open book with three binding components supporting the standard Stein fillable contact structure on T 3 (Lemma 1 and Proposition 2; also [34]). The small topology of these fillings will then allow us to show that many of these contact 3-manifolds admit symplectic Calabi-Yau caps introduced in [24], so as to conclude that they can not admit arbitrarily large Stein fillings (Theorem 7).…”

On the minimal number of singular fibers in Lefschetz fibrations over the torus

“…The value of N (g, h) has already been computed except N (g, 0) for g ≥ 3, N (g, 1) for g ≥ 2 and N (2,2). Recently Baykur and Korkmaz [1] found an interesting relation in the mapping class group M 1 2 of the genus-2 surface with one boundary component, and by using this relation they showed that N (2, 0) = 7. Furthermore, using the 8-holed torus relation [15] and the Matsumoto-Cadavid-Korkmaz relation [3,13,19], Hamada [12] found an upper bound for N (g, 1): he showed that N (g, 1) ≤ 4 if g ≥ 5 and N (g, 1) ≤ 6 if g = 3, 4.…”