arXiv:2611.14771v1 Announce Type: new Abstract: We give three fundamental examples highlighting the relationship between loops of spheres in $\#^k S^2\times S^2$, pseudo-isotopies, and diffeomorphisms of the 4-sphere. First, we describe an explicit pseudo-isotopy and Hatcher-Wagoner family that realizes the barbell diffeomorphism $\beta$ of the 4-sphere by constructing a finger-Whitney system for the family. From the Hatcher-Wagoner family, we are able to conclude that the example in \S 7 of ``Pseudo-isotopies of simply connected 4-manifolds'' generates $\beta$. Next, it is known that the square of $\beta$ is isotopic to id. We reprove this by building an explicit 2-parameter family of Morse functions that trivializes the Hatcher-Wagoner family for $\beta^2$. We do this by utilizing exchange points and families of handle slides. In the final example, we show that another Hatcher-Wagoner family, affectionately called the double cross, also generates $\beta$.
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