Geography of symplectic 4- and 6-manifolds

Journal of the Australian Mathematical Society, 2011

In this paper we study the geography and botany of symplectic spin 4-manifolds with abelian fundamental group. Building on the constructions in [19] and [21], the techniques employed allow us to give alternative proofs and extend their results to the nonsimply connected realm.

Algebraic & Geometric Topology, 2005

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integer called the degeneracy. In this paper we include the degeneracy as an extra parameter in the geography problem and show how to fill out the geography of symplectic 4-manifolds with Kodaira dimension 1 for all admissible triples.

Topology and its Applications, 2010

In this paper we construct several irreducible 4-manifolds, both small and arbitrarily large, with abelian non-cyclic fundamental group. The manufacturing procedure allows us to fill in numerous points in the geography plane of symplectic manifolds with π 1 = Z ⊕ Z, Z ⊕ Z p and Z q ⊕ Z p (gcd(p, q) = 1). We then study the botany of these points for π 1 = Z p ⊕ Z p .

arXiv (Cornell University), 2001

In this paper we are interested in the fundamental groups of closed symplectically aspherical manifolds. We apply the Donaldson theorem on hyperplane sections of closed symplectic manifolds in order to get new examples of such groups. Motivated by some results of Gompf, we introduce two classes of fundamental groups $\pi_1(M)$ of symplectically aspherical manifolds $M$ with $\pi_2(M)=0$ and $\pi_2(M)\not=0$. Relations between these classes are discussed. We show that several important classes of groups can be realized in both classes. Also, we notice that there are some dimensional phenomena in the realization problem.

2000

The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds.

Mathematische Zeitschrift, 2004

A smooth manifold M is called symplectically aspherical if it admits a symplectic form ω with ω|π 2(M) = 0. It is easy to see that, unlike in the case of closed symplectic manifolds, not every finitely presented group can be realized as the fundamental group of a closed symplectically aspherical manifold. The goal of the paper is to study the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups π 1(M) of symplectically aspherical manifolds M. The first one consists of fundamental groups of such M with π 2(M)=0, while the second with π 2(M)≠0. Relations between these classes are discussed. We show that several important (classes of) groups can be realized in both classes, while some groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general study of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical.

Communications in Contemporary Mathematics, 2007

We study the number of Darboux charts needed to cover a closed connected symplectic manifold (M, ω) and effectively estimate this number from below and from above in terms of the Lusternik–Schnirelmann category of M and the Gromov width of (M, ω).

Journal of the Korean Mathematical Society

For a closed symplectic 4-manifold X, let Diff 0 (X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of Diff 0 (X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {n 1 , n 2 ,. .. , n k } and any non-negative integer m, there exists a closed symplectic (or Kähler) 4-manifold X with b + 2 (X) > m such that the homologies H i of the quotient space Diff 0 (X)/Symp(X) over the rational coefficients are non-trivial for all odd degrees i = 2n 1 − 1,. .. , 2n k − 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Matić.

Third International Congress of Chinese Mathematicians, 2008

Asian Journal of Mathematics, 2002

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least -1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least -1 in any 4-manifold admitting a symplectic structure.