Minimal Symplectic 4-manifold Research Articles

Slope inequalities for the geography problem of nonspin symplectic 4-manifolds

Given an ordered pair of positive integers (e,σ), we can ask whether the pair is realizable as the Euler characteristic and the signature of a closed simply connected nonspin minimal symplectic 4-manifold. It is well known that for any fixed signature value σ>0, the pair is realizable if the Euler characteristic value e is large enough relative to σ. In this paper, we discuss just how large e might have to be asymptotically as we let σ→∞.

Small Lefschetz fibrations and exotic 4-manifolds

We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces $${\mathbb {CP}}^{2}\# p\, \overline{\mathbb {CP}}^{2}$$ for $$p=7, 8, 9$$ , and to $$3 {\mathbb {CP}}^{2}\#q\, \overline{\mathbb {CP}}^{2}$$ for $$q =12, \ldots ,19$$ . 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 $$b^+ \le 3$$ , 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 $${\mathbb {CP}}^{2}\# 4 \overline{\mathbb {CP}}^{2}$$ and $$3 {\mathbb {CP}}^{2}\# 6 \overline{\mathbb {CP}}^{2}$$ from small Lefschetz fibrations over surfaces of higher genera.

Minimality and fiber sum decompositions of Lefschetz fibrations

We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g > 1 Lefschetz fibrations on minimal symplectic 4-manifolds which, up to diffeomorphisms of the summands, admit unique decompositions as fiber sums.

New exotic 4-manifolds via Luttinger surgery on Lefschetz fibrations

In [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794], the first author constructed the first known example of exotic minimal symplectic[Formula: see text] and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to [Formula: see text]. The construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] uses Yukio Matsumoto's genus two Lefschetz fibrations on [Formula: see text] over 𝕊2 along with the fake symplectic 𝕊2 × 𝕊2 construction given in [Construction of symplectic cohomology 𝕊2 × 𝕊2, Proc. Gökova Geom. Topol. Conf.14 (2007) 36–48]. The main goal in this paper is to generalize the construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz and Yusuf Gurtas on [Formula: see text] for any k ≥ 2 and n = 1, and k ≥ 1 and n ≥ 2, respectively. Using our symplectic building blocks, we also construct new symplectic 4-manifolds with the free group of rank s ≥ 1, the free product of the finite cyclic groups, and various other finitely generated groups as the fundamental group.

Simply connected minimal symplectic 4-manifolds with signature less than −1

For each pair $(e,\sigma)$ of integers satisfying $2e+3\sigma\ge 0$, $\sigma\leq -2$, and $e+\sigma\equiv 0\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $\sigma$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).

Constructions of small symplectic 4-manifolds using Luttinger surgery

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#3(-CP^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#5(-CP^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.

Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds

In this article we construct an infinite family of simply connected minimal symplectic 4-manifolds, each of which admits at least two nonisomorphic Lefschetz fibration structures with the same generic fiber. We obtain such examples by performing knot surgery on an elliptic surface E(n) using a special type of 2-bridge knots.

Symplectic 4-manifolds with arbitrary fundamental group near the Bogomolov--Miyaoka--Yau line

In this paper we construct a family of symplectic 4--manifolds with positive signature for any given fundamental group $G$ that approaches the BMY line. The family is used to show that one cannot hope to do better than than the BMY inequality in finding a lower bound for the function $f=\chi+b\sigma$ on the class of all minimal symplectic 4-manifolds with a given fundamental group.

Einstein Metrics, Symplectic Minimality, and Pseudo-Holomorphic Curves

Let (M4, g,ω) be a compact, almost-Kahler–Einstein manifold of negative star-scalar curvature. Then (M,ω) is a minimal symplectic 4-manifold of general type. In particular, M cannot be differentiably decomposed as a connected sum $$N\,{\#}\,\overline{\mathbb{CP}}_2$$ .

Einstein Metrics, Symplectic Minimality, and Pseudo-Holomorphic Curves

Let (M, g, omega) be a compact, almost-Kaehler Einstein 4-manifold of negative star-scalar curvature. Then (M, omega) is a MINIMAL symplectic 4-manifold of general type. In particular, M cannot be differentiably decomposed as a connected sum N # (-CP_2).

NON-COMPLEX SYMPLECTIC 4-MANIFOLDS WITH $\lowercase{b}_{2}^+ =1$

Simply connected closed symplectic 4-manifolds with b 2 + = 1 and K2 = 0 are investigated. As a result, it is confirmed that most of homotopy elliptic surfaces {E(1)k|K is a fibred knot in S3} constructed by R. Fintushel and R. Stern in Invent. Math. 134 (1998) 363–400 are simply connected closed minimal symplectic 4-manifolds that do not admit a complex structure. 2000 Mathematics Subject Classification 57R17, 57R57 (primary), 14J26 (secondary).