Narrow Results

In [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794], the first author constructed the first known example of exotic minimal symplectic and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to . The construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] uses Yukio Matsumoto's genus two Lefschetz fibrations on 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 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.

In this paper, we examine mapping class group relations of some symplectic manifolds. For each $n\ge 1$ and $k\ge 1$, we show that the $2n$-dimensional Weinstein domain $W=\{f=\delta \}\cap B^{2n+2}$, determined by the degree $k$ homogeneous polynomial $f\in ℂ[z_0,\dots ,z_n]$, has a Boothby–Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case $n=2$ recovering the classical chain relation for the torus with two boundary components.

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus two fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to ${ℂ\mathbb{P}}^2\#6{\overline{ℂ\mathbb{P}}}^2$, ${ℂ\mathbb{P}}^2\#7{\overline{ℂ\mathbb{P}}}^2$, and $3{ℂ\mathbb{P}}^2\#k{\overline{ℂ\mathbb{P}}}^2$ for $k=16,17,18,19$ using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using $g$-nodal spherical deformations of certain singular fibers of genus $g\ge 3$ fibrations.

Suppose that $\Sigma$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of $\Sigma$, having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $\Sigma$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

We examine some questions regarding which torus links admit colorings by elements of a Dehn quandle.

We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local version of Harer’s twisting, which is presented in two different (but stably equivalent) forms. Our approach relies on 4-dimensional Lefschetz fibrations, and on 3-dimensional contact topology, via the Giroux-Goodman stable equivalence theorem for open book decompositions representing homologous contact structures.

In this paper, we introduce the notion of Lefschetz open book decompositions for closed oriented smooth $4$-manifolds. We define the notion of a Murasugi sum of Lefschetz open book decompositions. We show that the closed oriented $4$-manifold associated to a Murasugi sum of Lefschetz open books of two closed oriented $4$-manifolds $X_1$ and $X_2$ is the connected sum of $X_1$ and $X_2$. We also introduce the notion of a stabilization of a Lefschetz open book. We introduce the notion of a Lefschetz fibered link in a closed oriented $4$-manifold and using the stabilization of Lefschetz open book, we show that in a simply connected closed oriented $4$-manifold $X$, every oriented link $B$ in $X$ is Lefschetz fibered. We also discuss the existence of a folded Kähler structure on a closed oriented $4$-manifold associated to a given Lefschetz open book which essentially follows by the work of Baykur.

Loi–Piergallini and Akbulut–Ozbagci showed that every compact Stein surface admits a Lefschetz fibration over the disk D² with bounded fibers. In this note we give a more intrinsic alternative proof of this result.

For any $g\ge 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes of all Lefschetz fibrations over the two-sphere are not realized as slopes.

We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fiber. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.

We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which is formulated as an LF-structure on a shadowed polyhedron. We will show that the shadowed polyhedron constructed from a divide satisfies this property and the Lefschetz fibration of this polyhedron is isomorphic to the Lefschetz fibration of the divide. Furthermore, applying the same technique to certain free divides we will show that the links of those free divides are fibered with positive monodromy.