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