AI 中文总结
该研究证明特定Lidman-Piccirillo片段$V$单连通，进而构造出与$S^2\times S^2$、$\mathbb{CP}^2\\#\overline{\mathbb{CP}}^2$同胚但不微分同胚的奇异4维流形，还得到同胚却切片性不同的$B$与$W$
AI 中文摘要
我们证明，特定的Lidman-Piccirillo片段$V$——这是一个辛4维流形，具有$S^2\times D^2$的同调，由亏格为2的曲面在一次穿孔环面上的纤维丛经两次Luttinger手术构造而成——对于两次手术参数化的明确允许选择，它是单连通的。由此得出三个推论：1. 辛双曲$Z=V\cup_\sigma V$与$S^2\times S^2$同胚但不微分同胚；2. Lidman-Piccirillo流形$B$与$W$同胚，由于八字结在$B$中是切片的而在$W$中不是，它们是第一对由无约束的纽结切片性区分的同胚闭4维流形，即切片性对切片盘的同调类无约束，通过这种方式检测光滑结构可追溯至Casson的工作；3. 将Lidman和Piccirillo的定理2重新应用于$Z$，得到一个闭单连通4维流形，它与$\mathbb{CP}^2\\#\overline{\mathbb{CP}}^2$同胚但不微分同胚。这些推论源于单连通性结论，结合Freedman分类、Hambleton-Kreck分类以及手术参数的刚性分析。基本群以Baldridge和Kreck的风格计算，从每条经线的明确基于代表元与拉格朗日推前得到，所得关系系统经陪集计数判定，且通过两个答案已知的构型校准。推导计算与有限表现判定可从辅助文件复现。
英文摘要
We prove that a specified Lidman-Piccirillo piece $V$, a symplectic $4$-manifold with the homology of $S^2\times D^2$ built from a genus-$2$ surface bundle over a once-punctured torus by two Luttinger surgeries, is simply connected, for an explicit permitted choice of the two surgery parametrizations. Three consequences follow. The symplectic double $Z=V\cup_σV$ is homeomorphic but not diffeomorphic to $S^2\times S^2$. The Lidman-Piccirillo manifolds $B$ and $W$ are homeomorphic. Since the figure-eight knot is slice in $B$ and not in $W$, they are the first pair of homeomorphic closed $4$-manifolds distinguished by unconstrained knot slicing, that is, by sliceness with no constraint on the homology class of the slice disk; detecting smooth structure this way goes back to Casson. Finally, the regluing of Lidman and Piccirillo's Theorem~2 applied to $Z$ yields a closed simply connected $4$-manifold homeomorphic but not diffeomorphic to $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. The consequences follow from the simple-connectivity statement by the classifications of Freedman and of Hambleton-Kreck, together with a rigidity analysis of the surgery parameters. The fundamental group is computed in the style of Baldridge and Kirk, from explicit based representatives of every meridian and Lagrangian push off, and the resulting relation system is decided by coset enumeration, after calibration on two configurations whose answers are known independently. The development calculations and finite-presentation decisions can be reproduced from the ancillary files.
Comments32 pages, 2 figures
