普通与等变Dehn手术数之间的无界间隙
Unbounded Gaps Between Ordinary and Equivariant Dehn Surgery Numbers
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中文总结 AI 辅助
该研究证明对每个整数k≥1，存在一对闭定向3维流形与对合使得其普通Dehn手术数为k、等变Dehn手术数为2k，差值无界，还构造了满足DS(Z)<EDS(Z,σ)的无限多互不同胚不可约透镜空间，解答了K3问题列表的相关问题。
中文摘要 AI 辅助
对于闭定向3维流形Y及保定向对合τ，令DS(Y)表示Y的整数手术描述的最小分支数，EDS(Y,τ)表示诱导τ的周期手术描述的对应最小值。我们证明对每个整数k≥1，存在一对(Y_k,τ_k)使得DS(Y_k)=k，EDS(Y_k,τ_k)=2k。因此，即使τ为对合，EDS(Y,τ)-DS(Y)的差值也无界。这解答了K3问题列表中的问题1.15(b)和1.15(c)。我们还构造了无限多个互不同胚的不可约透镜空间Z，其存在对合σ满足DS(Z)<EDS(Z,σ)。
英文摘要
For a closed oriented $3$-manifold $Y$ and an orientation-preserving involution $τ$, let $\DS(Y)$ denote the minimum number of components in an integral surgery description of $Y$, and let $\EDS(Y,τ)$ denote the corresponding minimum among periodic surgery descriptions inducing $τ$. We prove that for every integer $k\geq 1$ there is a pair $(Y_k,τ_k)$ such that \[ \DS(Y_k)=k, \qquad \EDS(Y_k,τ_k)=2k. \] Consequently, the difference $\EDS(Y,τ)-\DS(Y)$ is unbounded even when $τ$ is an involution. This answers Problems~1.15(b) and~1.15(c) in the K3 problem list. We also construct infinitely many pairwise nonhomeomorphic irreducible lens spaces $Z$ admitting involutions $σ$ for which \[ \DS(Z)<\EDS(Z,σ). \]
