For each pair (e, σ) of integers satisfying 2e + 3σ ≥ 0, σ ≤ −2, and e + σ ≡ 0 (mod 4), with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic e and signature σ. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature −1) with Euler characteristic 4k (resp. 4k + 1) for all k ≥ 46 (resp. k ≥ 49).
Abstract. In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP 2 #3CP 2 which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to 3CP 2 #5CP 2 which has two essential Lagrangian tori with simply connected complement. These manifolds can be used to replace E(1) in many known theorems and constructions. Examples in this article include the smallest known minimal symplectic manifolds with abelian fundamental groups including symplectic manifolds with finite and infinite cyclic fundamental group and Euler characteristic 6.
The main results of this paper describes a formula for the Seiberg–Witten invariant of a 4-manifold which admits a nontrivial free circle action. We use this theorem to produce a nonsymplectic 4-manifold with a free circle action whose orbit space fibers over circle. We also describe a nontrivial 3-manifold which is not the orbit space of any symplectic 4-manifold with a free circle action. A corollary of the main theorem is a formula for the 3-dimensional Seiberg–Witten invariants of the total space of a circle bundle over a surface.
In this article we study the problem of minimizing aχ + bσ on the class of all symplectic 4-manifolds with prescribed fundamental group G (χ is the Euler characteristic, σ is the signature, and a, b ∈ R), focusing on the important cases χ, χ + σ and 2χ + 3σ. In certain situations we can derive lower bounds for these functions and describe symplectic 4-manifolds which are minimizers. We derive an upper bound for the minimum of χ and χ + σ in terms of the presentation of G.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.