AI 中文总结
本文研究四流形中由外围同态控制的环面外部替换，构造光滑同伦4-球面及与$S^2\times S^2$等胚流形，并分析粘合产生的有限非阿贝尔基本群。
AI 中文摘要
我们研究四流形中的环面外部替换，其构造由外围同态控制。利用 Boyle 以及 Kanenobu-Kazama 的工作中产生的环面外部，并显式控制其基本群和外围系统，我们构造了光滑同伦 4-球面以及与 $S^2 \times S^2$ 和 $\\#_3(S^2 \times S^2)$ 同胚的流形。我们还研究两个环面外部的直接粘合。两个单零外部的交换粘合会消去两个子午线，并给出欧拉示性数为 $4$、符号差为 $0$ 的单连通流形，而双零外部的保子午线粘合则将补空间群作为子午线子群上的融合积保留下来。扭曲的外围粘合给出循环的和显式的有限非阿贝尔基本群。我们还考虑了完整的 Kanenobu-Kazama $1$-柄族以及具有满秩外围子群的 Litherland 环面；后者没有零原始边界斜率，并自然导致融合积基本群。
英文摘要
We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kanenobu-Kazama, with explicitly controlled fundamental groups and peripheral systems, we construct smooth homotopy 4-spheres and manifolds homeomorphic to S^2 x S^2 and \#_3(S^2 x S^2). We also study direct gluings of two torus exteriors. A swap gluing of two one-null exteriors kills both meridians and gives a simply connected manifold with Euler characteristic \(4\) and signature \(0\), while a meridian-preserving gluing of doubly-null exteriors retains the complement groups as an amalgam over the meridian subgroup. Twisted peripheral gluings give cyclic and explicit finite nonabelian fundamental groups. We also consider the full Kanenobu-Kazama \(1\)-handle family and Litherland tori with full-rank peripheral subgroup; the latter have no null primitive boundary slope and lead naturally to amalgamated-product fundamental groups.
Comments29 pages
