Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery

Abstract: Watanabe constructed many nontrivial family diffeomorphisms in $π<em>*\text{Diff}</em>\partial(D<sup>4)$, 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 $π<em>{k-1}\text{Diff}</em>\partial(M_{k+1}&#39;)$ with $M_{k+1}&#39;=\natural_{k+1} S^2\times D<sup>2$ 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 $D<sup>4$ 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.