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

David Gabai, David T. Gay, Daniel Hartman

TL;DR

This work develops codimension-2 techniques for simply connected 4-manifolds to detect nontrivial loops of embedded 2-spheres, culminating in a $ extbf{I}$-invariant valued in $\mathbb{Z}_2^k$ for loops in $\#^k(S^2\times S^2)$. Central to the construction are finger/Whitney disc data, immersion- versus embedded-arc normalization (IA/EA), and a robust scheme of disc slides and switching that yields a well-defined invariant independent of choices. The paper builds a two-tier framework: first a one-parameter, finger-first reduction that renders the invariant computable, then a comprehensive two-parameter theory controlling all possible deformations of the data via five moves (disc/sphere slides, births/deaths, $x^3$-moves, saddles) and FW-vector fields. The results lay the groundwork for applying pseudo-isotopy theory to 4-manifolds, with the long-term aim of proving that $\text{Diff}^+(S^4)$ contains exotic elements, as developed in the sequel GGH. Overall, the work provides a detailed, parity-based codimension-2 framework for distinguishing loops of embedded spheres that are not realized by standard embedding-space loops and links this to potential diffeomorphism obstructions in dimension four.

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.

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

TL;DR

This work develops codimension-2 techniques for simply connected 4-manifolds to detect nontrivial loops of embedded 2-spheres, culminating in a -invariant valued in for loops in . Central to the construction are finger/Whitney disc data, immersion- versus embedded-arc normalization (IA/EA), and a robust scheme of disc slides and switching that yields a well-defined invariant independent of choices. The paper builds a two-tier framework: first a one-parameter, finger-first reduction that renders the invariant computable, then a comprehensive two-parameter theory controlling all possible deformations of the data via five moves (disc/sphere slides, births/deaths, -moves, saddles) and FW-vector fields. The results lay the groundwork for applying pseudo-isotopy theory to 4-manifolds, with the long-term aim of proving that contains exotic elements, as developed in the sequel GGH. Overall, the work provides a detailed, parity-based codimension-2 framework for distinguishing loops of embedded spheres that are not realized by standard embedding-space loops and links this to potential diffeomorphism obstructions in dimension four.

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 and in connected sums of . 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 has an exotic element.
Paper Structure (32 sections, 79 theorems, 17 equations, 55 figures, 3 tables)

This paper contains 32 sections, 79 theorems, 17 equations, 55 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Immersed arc position
  4. Finger first position
  5. Independence of ordering
  6. Loops of multi-spheres
  7. Introduction
  8. Immersed arc position
  9. Finger first position
  10. Independence of ordering
  11. 2-parameter families of embeddings
  12. Generic families of embeddings.
  13. Local analysis of paths of embeddings
  14. Local finger-Whitney data
  15. Local analysis of 2-parameter families.
  16. ...and 17 more sections

Key Result

Theorem 3

For each $k$ there exists a surjective homomorphism with $\pi_1(\mathop{\mathrm{LB}}\nolimits,\mathcal{R}^{\mathop{\mathrm{std}}\nolimits})$ contained in the kernel of $\mathop{\mathrm{\textbf{I}}}\nolimits$. Furthermore, if $\mathop{\mathrm{\textbf{I}}}\nolimits([\alpha]) = (x_1, \ldots, x_k)$ and we extend $\alpha$ to a loop $\alpha'$ in $\mathop{\ In particular, for all $k$, there are homotopi

Figures (55)

  • Figure 1: The Key Example
  • Figure 2: $G$-Disc Sliding $w_i$ over $w_j$ along $\omega$
  • Figure 3: Constructing a Twisted $G$-Whitney Disc Slide
  • Figure 4: Going from Immersed Arc to Embedded Arc Position
  • Figure 5: Framed Finger Form
  • ...and 50 more figures

Theorems & Definitions (209)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Remark 10
  • Theorem 11
  • Remark 1.1
  • ...and 199 more