Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds

Abstract

Abstract. We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twists, answering a problem posed by Auroux.

Small Lefschetz fibrations and exotic 4-manifolds

“…. , 8, we obtain genus-2 Lefschetz fibrations of the types (12,9), (14,8), (16,7), (18,6), (20,5), (22,4), (24,3), (26,2). They are all minimal by Proposition 6, and once we show that X i is simply-connected, we can once use Theorem 13 again to conclude that X i is an exotic 3CP 2 #(11 + i)CP 2 , for i = 1, .…”

“…For each i = 1, 2, 3, let ( X i , f i ) be the Lefschetz fibration prescribed by the positive factorization W i , so that we have genus-2 Lefschetz fibrations of types of (8,6), (10,5) and (12,4). By Proposition 6, each one of them is minimal.…”

“…We note that when φ is a Dehn twist t ±1 α , the symplectic Lefschetz fibration 5,12]. This is a special instance of Luttinger surgery, where one cuts out a Weinstein tubular neighborhood of an embedded Lagrangian torus T in a symplectic 4-manifold X and glues it back in differently in a way that the symplectic form in the complement extends [35,6].…”

“…Lastly, for n + 2s = 30, the pairs (2,14), (4,13), (6,12) are ruled out by the second inequality in Lemma 5, and (28, 1), (30, 0) by (22).…”

“…We are thus left with the possible values (8,6), (10,5), (12,4) when b + = 1 and (10, 10), (12,9), (14,8), (16, 7), (18,6), (20,5), (22,4), (24,3), (26,2) when b + = 3. In all these cases s > 0, so the presence of a reducible fiber implies that there exists a homology class with an odd intersection number.…”