Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.01833

Braided Multisections and Aspherical Stein Fillings

Anar Akhmedov
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct Stein fillings diffeomorphic to complements of a connected symplectic unbranched multisection and a regular sphere fiber in the ruled surface $Σ_g\times S^2$. For every $g\geq2$ and every multisection degree $p\geq4$, a fixed contact three-manifold admits infinitely many pairwise nonhomeomorphic aspherical fillings of this form. Their first and second integral homology groups are $\Z^{2g}\oplus\Z/p$ and $\Z^{(2g-1)(p-1)}$, respectively, and their intersection pairings vanish. We vary the braid monodromy while preserving the marked contact boundary. Unbounded torsion orders in the first homology of regular unbranched covers of degree $p2^{p-1}$ detect infinitely many isomorphism types of fundamental groups. We also give an equivariant Weinstein handle construction for the cyclic filling $W_{g,4}$. The contact boundaries are supported by spinal open books with planar pages and have positive ordinary support genus.

Explore related subjects

math.GTmath.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Anar Akhmedov. 2026-11-01. Braided Multisections and Aspherical Stein Fillings. https://arxiv.org/abs/2611.01833

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

2-Linearizability of Geometric 3-Manifold Groups Over Commutative Rings

The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $\mathrm{PGL}(2,R)$, where $R$ is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but $\mathbb{S}^3$ and $\widetilde{\mathrm{SL}_2}$. In the case of $\mathbb{S}^3$, the conjecture holds if we replace $\mathrm{PGL}(2,R)$ with $\mathrm{GL}(2,R)$. A spherical counterexample for the projective variant is the Poincaré homology sphere $Σ(2,3,5)$. In the case of $\widetilde{\mathrm{SL}_2}$, the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere $Σ(2,3,7)$.

math.GT↗
arXiv2026

Erratum to "Linear representations of 3-manifold groups over rings"

The fundamental groups of compact $3$-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by allowing only finite groups of the form $\mathrm{PGL}_2(R),$ where $R$ is a finite commutative ring with identity. In a prior paper, "Linear representations of $3$-manifold groups over rings", we, together with two co-authors, gave an equivalent formulation of Luo's conjecture via faithful representations, investigated variants of the conjecture where the target group is of type $\mathrm{SL}_2, \mathrm{GL}_2, \mathrm{PSL}_2$, or $\mathrm{PGL}_2$ as in the original conjecture, and claimed to provide a $3$-manifold group which violated all four variants of the conjecture. We have discovered a flaw in the work, such that the existing proof is valid only for $\mathrm{SL}_2$ and $\mathrm{GL}_2$. Here, we provide a modified argument for the $\mathrm{PSL}_2$ and $\mathrm{PGL}_2$ cases.

math.GT↗
arXiv2026

Universal bounds for decompositions of PL manifolds

The geometric dimension of a nonempty compact PL manifold $N$ is the smallest nonnegative integer $d$ for which $N$ admits a handle decomposition with all handles of index at most $d$. A PL filling of a compact PL manifold is a decomposition into compact codimension $0$ PL submanifolds, called sectors, with pairwise disjoint interiors and intersections described by standard local models. We advance Singhof's program of simplifying the sectors of a filling and their intersections. Specifically, for integers $m, \ell\ge 1$ and nonnegative integers $a_1\le\cdots\le a_\ell$, we prove that every closed PL $m$-manifold admits a filling into $\ell$ sectors whose nonempty $r$-fold intersections have geometric dimension at most $a_r$ for each $1\le r\le \ell$ if and only if \[ a_{λ_1}+\cdots+a_{λ_q}\ge m-\ell+1 \] for every partition $λ_1+\cdots+λ_q=\ell$ into positive integers. Under the same inequalities, any prescribed $\ell$-sector filling of the boundary of a compact PL $m$-manifold extends to a filling with the corresponding relative handle index bounds. Necessity follows from the cup-product obstructions of Courte, Moussard, Ren, and Zhou. To prove sufficiency, we develop compatible handle trades that simplify intersections in increasing order of depth. For nondecreasing tuples of nonnegative integers, our theorem resolves the PL classification problem posed by Courte, Moussard, Ren, and Zhou. In particular, we obtain the bound $(2,2,4)$ for $3$-sector fillings in dimension $6$ and improve the $5$-sector bound $(1,2,3,3,4)$ of Rubinstein and Tillmann to $(1,2,2,3,4)$ in dimension $8$.

math.GT↗