Xiayu Tan | Search arXiv

arXiv2026

Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery

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.

math.GT↗

arXiv2026

Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps

For a half-unknotted implanted $(i,n-i)$-barbell $β=β_{i,n-i}$ in $M^n$, we construct two specific pseudo-isotopies, which we denote by standard barbell pseudo-isotopies, both resulting in that barbell diffeomorphism, each having a Cerf diagram only containing a single eye and with easily computable Hatcher-Wagoner invariants. We give an explicit formula for $β_{2,n-2}$ and a special class of $β_{3,n-3}$. Using this we show that for $n\geq 6$, every pseudo-isotopy with vanishing first Hatcher-Wagoner invariant can be isotoped to a composition of standard barbell pseudo-isotopies with $i=2$ or $3$. In dimension $n=4$, we further generalize the constructions and computations to half-unknotted immersed barbell diffeomorphisms and prove that for every $s\in \mathbb{Z}_2, σ\in π_2 M,γ\in π_1 M$ with $s=0 \text{ or }w_2^M(σ)\neq0$, there exists a standard immersed barbell pseudo-isotopy $f_β$ with the second induced Hatcher-Wagoner invariant $Θ(f_β)=(s,σ)\cdot [γ]$.

math.GT↗