The Smoothability of Embedded Surfaces in 4-Manifolds Under Stabilizations
Abstract
We study the smooth stabilization number of a locally flat embedded surface in a smooth $4$-manifold, defined as the minimum number of connected sums with S2× S2 needed to make the surface topologically isotopic to a smooth embedding. We prove that a nonsmoothable locally flat surface in a closed, simply connected smooth $4$-manifold becomes smoothable after a single stabilization if its homology class is primitive and noncharacteristic and its complement has abelian fundamental group. This yields infinitely many examples of nonsmoothable surfaces with smooth stabilization number one. We also exhibit examples for which a single stabilization produces infinitely many pairwise smoothly nonisotopic representatives in the same topological isotopy class. In contrast, we construct a locally flat genus-eight surface representing 6h in ℂℙ2 whose smooth stabilization number is exactly two. More generally, we show that smooth stabilization numbers of locally flat embedded surfaces in ℂℙ2 are unbounded.
Related
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- Embedded minimal surfaces with three ends associated with the Hoffman-Osserman catenoids
- Minimal surfaces in ℍ2×ℝ with finite total curvature and embedded ends
- Smoothing topological isotopy of surfaces in 4-manifolds with a boundary geometric dual
- Links and singularities of stable maps from $3$-manifolds to surfaces
BibTeX
@misc{zhang2026smoothability,
title = {{The Smoothability of Embedded Surfaces in 4-Manifolds Under Stabilizations}},
author = {Zuyi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.24834},
eprint = {2609.24834},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24834},
}
RIS
TY - UNPB
TI - The Smoothability of Embedded Surfaces in 4-Manifolds Under Stabilizations
AU - Zuyi Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.24834
UR - https://arxiv.org/abs/2609.24834
AB - We study the smooth stabilization number of a locally flat embedded surface in a smooth $4$-manifold, defined as the minimum number of connected sums with S^2× S^2 needed to make the surface topologically isotopic to a smooth embedding. We prove that a nonsmoothable locally flat surface in a closed, simply connected smooth $4$-manifold becomes smoothable after a single stabilization if its homology class is primitive and noncharacteristic and its complement has abelian fundamental group. This yields infinitely many examples of nonsmoothable surfaces with smooth stabilization number one. We also exhibit examples for which a single stabilization produces infinitely many pairwise smoothly nonisotopic representatives in the same topological isotopy class. In contrast, we construct a locally flat genus-eight surface representing 6h in ℂℙ^2 whose smooth stabilization number is exactly two. More generally, we show that smooth stabilization numbers of locally flat embedded surfaces in ℂℙ^2 are unbounded.
ER -