Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917461918482432 |
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| author | Tan, Xiayu |
| author_facet | Tan, Xiayu |
| contents | 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 $Θ(f_β)=(s,σ)\cdot [γ]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_00939 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps Tan, Xiayu Geometric Topology 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 $Θ(f_β)=(s,σ)\cdot [γ]$. |
| title | Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2604.00939 |