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

TL;DR

This paper develops new methods in pseudo-isotopy and embedding space theory to detect nontrivial loops of embedded 2-spheres in certain 4-manifolds, revealing complex diffeomorphism structures.

Contribution

It introduces an invariant for detecting nontrivial loops of spheres in 4-manifolds, advancing understanding of diffeomorphisms of the 4-sphere.

Findings

Invariant detects nontrivial loops of spheres in $S^{2} imes S^{2}$

Shows existence of non-homotopic loops of embedded spheres

Prepares to prove the existence of exotic elements in $ ext{Diff}^+(S^4)$

Abstract

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 $\operatorname{Diff}^{+}(S^{4})$ has an exotic element.