4-流形中奇异曲面的内部与外部稳定化

An atomic approach to Wall-type stabilization problems

Wall-type stabilization problems investigate the collapse of exotic 4-dimensional phenomena under stabilization operations (e.g., taking connected sums with $S^2 \times S^2$). We propose an elementary approach to these problems, providing a construction of exotic 4-manifolds and knotted surfaces that are candidates to remain exotic after stabilization -- including examples in the setting of closed, simply connected 4-manifolds. As a proof of concept, we show this construction yields exotic surfaces in the 4-ball that remain exotic after (internal) stabilization, detected by the cobordism maps …

对于带边界的怪异曲面，一次稳定化是不够的

For Exotic Surfaces with Boundary, One Stabilization is Not Enough

该研究针对带边界怪异曲面单次稳定化是否足以实现光滑同痕的问题，通过Floer理论技术分析稳定化距离随卫星运算的变化，证明4维球中存在稳定化距离任意大的怪异圆盘，给出“一次稳定化不够”的首批例子。

Exotic codimension-1 submanifolds in 4-manifolds and stabilizations

In a small simply-connected closed 4-manifold, we construct infinitely many pairs of exotic codimension-$1$ submanifolds with diffeomorphic complements that remain exotic after any number of stabilizations by $ S^2 \times S^2$. We also give new constructions of exotic embeddings of 3-spheres in 4-manifolds with diffeomorphic complements.

一次稳定化对可缩4维流形而言是不够的

One stabilization is not enough for contractible 4-manifolds

该研究构造了与$S^2 \ imes S^2$取连通和后仍具异质性的cork例子，结合已有工作证明存在取该连通和后仍保持绝对异质性的可缩4维流形异质对，推进了4维流形稳定化相关研究。

Knotted surfaces in 4-manifolds and stabilizations

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabil…

一次稳定化对于闭纽结曲面是不够的

One stabilization is not enough for closed knotted surfaces

本文证明存在包含成对异构纽结2-球面的光滑4-流形，其异构性在一次稳定化后仍保持，表明“一次即足够”定理不适用于同调类为特征类的闭曲面。