AI 中文总结
本文推广杠铃微分同胚至高阶情形，构造非平凡族微分同胚，并证明Watanabe的钳夹手术构造可视为植入的高阶杠铃微分同胚，进而确立其非平凡性。
AI 中文摘要
Watanabe通过在三价图上进行钳夹手术构造了$\pi_*\text{Diff}_\partial(D^4)$中的许多非平凡族微分同胚。本文推广了Budney和Gabai发现的杠铃微分同胚，构造了$\pi_{k-1}\text{Diff}_\partial(M_{k+1}')$中的一系列非平凡元素，其中$M_{k+1}'=\natural_{k+1} S^2\times D^2$为$(k+1)$-环杠铃，我们称之为高阶杠铃微分同胚。我们证明，若三价图在图复形中同调非零，则Watanabe的构造可实现为$D^4$中的植入高阶杠铃微分同胚。结合该实现与Watanabe的检测定理，我们确立了这些植入高阶杠铃微分同胚的非平凡性。
英文摘要
Watanabe constructed many nontrivial family diffeomorphisms in $π_*\text{Diff}_\partial(D^4)$, which were obtained by doing clasper surgeries on trivalent graphs. In this paper, we generalize the barbell diffeomorphism discovered by Budney and Gabai and construct a series of nontrivial elements in $π_{k-1}\text{Diff}_\partial(M_{k+1}')$ with $M_{k+1}'=\natural_{k+1} S^2\times D^2$ the $(k+1)$-cuff barbell, which we call higher barbell diffeomorphisms. We show that Watanabe's constructions can be realized as implanted higher barbell diffeomorphisms in $D^4$ if the trivalent graph is homologically nonzero in the graph complex. Combining this realization with Watanabe's detection theorem, we establish the nontriviality of these implanted higher barbell diffeomorphisms.
Comments34 pages, 26 figures. Comments are very welcome!
