Nonsmoothable Embedded Surfaces in 4-Manifolds That Become Smoothable after a Single Stabilization

Zuyi Zhang

Home/Archive/arXiv

arXiv · 2609.24834

Nonsmoothable Embedded Surfaces in 4-Manifolds That Become Smoothable after a Single Stabilization

Zuyi Zhang

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we find infinitely many examples of nonsmoothable embedded surfaces in a 4-manifold that become smoothable after taking the connected sum with S^2 x S^2. To the author's knowledge, no such examples have previously been discovered. In [CK25], Cha and Kim asked for the minimal number of stabilizations required to turn a nonsmoothable surface into a smoothable one. We provide sufficient conditions ensuring that this number is 1. In addition, we show that there is a nonsmoothable surface in a K3 surface such that, after a single stabilization, its topological isotopy class contains infinitely many smooth representatives that are pairwise not smoothly isotopic.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Zuyi Zhang. 2026-09-21. Nonsmoothable Embedded Surfaces in 4-Manifolds That Become Smoothable after a Single Stabilization. https://arxiv.org/abs/2609.24834

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

KEEP EXPLORING

Related papers

arXiv2026

Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

We build a large family of hyperbolic links with an explicit decomposition of the complement into right-angled hyperbolic polytopes of finite volume. Namely, in a series of papers A.D.Mednykn and A.Yu.Vesnin introduced a construction that for a given right-angled polytope $P$ in geometry $\mathbb L^3$, $\mathbb R^3$, $\mathbb S^3$, $\mathbb L^2\times \mathbb R$, $\mathbb S^2\times \mathbb R$ and a Hamiltonian cycle, theta-subgraph or $K_4$-subgraph $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ has a complete hyperbolic structure of finite volume and generalize this criterion to similar links in $3$-manifolds different from $S^3$. We prove that hyperbolic links $C_Γ$ are parametrized by nonselfcrossing Eulerian cycles, Eulerian theta-subgraphs and Eulerian $K_4$-subgraphs in hyperbolic right-angled $3$-polytopes of finite volume in $\mathbb L^3$ with $0$, $2$ or $4$ finite vertices. The complement $S^3\setminus C_Γ$ is glued of $4$, $8$ or $16$ copies of the corresponding right-angled polytope. We give a criterion when the link $C_Γ$ consists of mutually unlinked circles and prove that if such a link is nontrivial, then it contains the Borromean rings. The latter problem is motivated by the Efimov effect in quantum mechanics.

math.GT↗

arXiv2026

Knots and the Sierpinski Tetrahedron

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.