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We prove a lower bound for the number of (nonseparating) vanishing cycles of a genus-g Lefschetz fibration. We also show that a fiber sum X#f X for a Lefschetz fibration X → S2 is a minimal symplectic 4-manifold. 1. Statement of results Assume that f : X → S is a relatively minimal genus-g Lefschetz fibration with s separating and n nonseparating vanishing cycles. (For definitions of the above notions see Section 2; the reader is also advised to turn to [GS] or [La] for further reference. In our subsequent disscussions we will always assume that f is injective on its critical set.) The fibration is called trivial if s = n = 0, in this case X is diffeomorphic to S × Σg hence it also admits a ruling over the genus-g surface Σg. In the following we will prove a bound for n: Theorem 1.1. If f : X → S is as above and the fibration is nontrivial then

On the number of vanishing cycles in Lefschetz fibrations