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Pseudo-isotopies of simply connected 4-manifolds

David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, Mark Powell

TL;DR

The paper completes Quinn's program for the 4-dimensional pseudo-isotopy theorems by supplying a replacement-criterion-free argument that yields both the topological and stable smooth pseud-isotopy results for simply connected 4-manifolds. It preserves the core Cerf-theoretic framework (nested eyes, Morse data) but replaces the problematic replacement step with a factorisation and dual-sphere stabilization strategy, coupled with a refined sum-square move to control intersections. A decomposition theorem for topological 4-manifolds with boundary is established, enabling Perron-type arguments in the topological setting and yielding a robust proof of the topological pseudo-isotopy theorem, along with a stable smooth analogue. The work also clarifies the status of the disc replacement criterion as an open problem and discusses its implications via an explicit S^4 diffeomorphism scenario, linking smoothing theory to deep questions about isotopy classes of homeomorphisms.

Abstract

Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that $π_4(\operatorname{TOP}(4)/\operatorname{O}(4)) =0$. A replacement criterion is used at a key juncture in Quinn's proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn's proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem.

Pseudo-isotopies of simply connected 4-manifolds

TL;DR

The paper completes Quinn's program for the 4-dimensional pseudo-isotopy theorems by supplying a replacement-criterion-free argument that yields both the topological and stable smooth pseud-isotopy results for simply connected 4-manifolds. It preserves the core Cerf-theoretic framework (nested eyes, Morse data) but replaces the problematic replacement step with a factorisation and dual-sphere stabilization strategy, coupled with a refined sum-square move to control intersections. A decomposition theorem for topological 4-manifolds with boundary is established, enabling Perron-type arguments in the topological setting and yielding a robust proof of the topological pseudo-isotopy theorem, along with a stable smooth analogue. The work also clarifies the status of the disc replacement criterion as an open problem and discusses its implications via an explicit S^4 diffeomorphism scenario, linking smoothing theory to deep questions about isotopy classes of homeomorphisms.

Abstract

Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that . A replacement criterion is used at a key juncture in Quinn's proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn's proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem.
Paper Structure (14 sections, 14 theorems, 16 equations, 18 figures)

This paper contains 14 sections, 14 theorems, 16 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Stable isotopy
  3. Topological isotopy
  4. Classification of homeomorphisms up to isotopy
  5. Morse functions, nested eye Cerf graphics, and the start of Quinn's proof
  6. Whitney spheres and the sum square move
  7. Whitney spheres
  8. The sum square move
  9. The proof of the inductive step
  10. A decomposition theorem for topological 4-manifolds
  11. The topological pseudo-isotopy theorem
  12. The disc replacement criterion
  13. Description of the problem
  14. An interesting diffeomorphism of $S^4$

Key Result

Theorem 1.1

Let $M$ be a compact, smooth, simply-connected 4-manifold and let $F \colon M \times [0,1] \to M \times [0,1]$ be a smooth pseudo-isotopy. Then $F$ is stably isotopic to the identity.

Figures (18)

  • Figure 1: A family of nested eyes.
  • Figure 2: Finger and Whitney discs with boundaries forming an arc in $A_1$ and in $B_1$. The Whitney spheres from \ref{['subsection:whitney spheres']} corresponding to the finger move discs are also shown.
  • Figure 3: A description of the Whitney sphere $S_{V_k^{ij}}$ in $\mathbb{R}^3\times \mathbb{R}$.
  • Figure 4: The sum square move
  • Figure 5: A sequence of Cerf moves that switch the order of nesting of two concentric eyes.
  • ...and 13 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.3: Quinn
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 19 more