Minimizing Euler Characteristics of Symplectic Four-manifolds

We prove that the minimal Euler characteristic of a closed symplectic four-manifold with given fundamental group is often much larger than the minimal Euler characteristic of almost complex closed four-manifolds with the same fundamental group. In fact, the difference between the two is arbitrarily large for certain groups. It was first proved by Dehn [2] that every finitely presentable group Γ can be realized as the fundamental group of a closed oriented smooth four-manifold. Taking the minimum over the Euler characteristics of all such manifolds, one obtains an interesting numerical invariant q (Γ) of finitely presentable groups; see for example [4, 5, 7]. As mentioned in [7], there are geometric variants q(Γ) of this definition, obtained by minimizing the Euler characteristic only over those fourmanifolds with fundamental group Γ which carry a specified geometric structure. One trivially has q (Γ) ≤ q(Γ) for all geometric structures. Moreover, the inequality is often strict. For a simple example of a geometric invariant, consider almost complex fourmanifolds. Every finitely presentable group is the fundamental group of an almost complex four-manifold [6], but the minimal Euler characteristic over almost complex four-manifolds is strictly larger than q (Γ) for many Γ. Nevertheless, in this case it is easy to see that the difference between the smooth and geometric invariants is universally bounded independently of Γ; compare [6]. The purpose of this paper is to show that in the symplectic category this boundedness fails. Recall that Gompf [3] proved that every finitely presentable Γ can be realised as the fundamental group of a closed symplectic four-manifold. Thus we can define q (Γ) to be the minimal Euler characteristic of a closed symplectic four-manifold with fundamental group Γ. Then we have Theorem 1. For every c > 0 there exists a finitely presentable group Γ satisfying q (Γ) ≥ q (Γ) + c . Proof. We shall use the sequence Fr of free groups of rank r. It suffices to show that the difference q (Fr)− q (Fr) Received by the editors May 3, 2005. 2000 Mathematics Subject Classification. Primary 57M07, 57R17, 57R57. The author is grateful to P. Kirk for pointing out the question that is answered here. c ©2006 American Mathematical Society