Pseudo-Isotopy and Diffeomorphisms of the 4-Sphere I: Loops of Spheres

We introduce new methods in pseudo-isotopy and embedding space theory.

As an application we introduce an invariant that detects nontrivial loops of embedded 2-spheres in $S^{2} \times S^{2}$ and in connected sums of $S^{2} \times S^{2}$. that cannot be homotoped to loops of spheres dual to the standard horizontal spheres.

In the sequel [GGH], we will use these techniques to expand upon the applicability of the invariant and prove $\operatorname{Diff}^{+}(S^{4})$ has an exotic element.