Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps

For a half-unknotted implanted $(i,n-i)$-barbell $\beta ={\beta }_{i,n-i}$ 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 ${\beta }_{2,n-2}$ and a special class of ${\beta }_{3,n-3}$. Using this we show that for $n\ge 6$, 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 $s\in {\mathbb{Z}}_2,\sigma \in {\pi }_{2}M,\gamma \in {\pi }_{1}M$ with $s=0\text{ or }{w}_2^M(\sigma )\mathrm{\ne }0$, there exists a standard immersed barbell pseudo-isotopy $f_{\beta }$ with the second induced Hatcher-Wagoner invariant $\mathrm{\Theta }(f_{\beta })=(s,\sigma )\cdot [\gamma ]$.