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The Smale conjecture and min-max theory

Daniel Ketover, Yevgeny Liokumovich

TL;DR

This work gives a new, min-max–based proof of the Smale conjecture for lens spaces, including RP^3, by reducing the problem to the homotopy type of spaces of Heegaard tori and then retracting to minimal Clifford tori. The authors classify minimal tori in lens spaces (via Clifford tori), control their parameter spaces with Goeritz groups, and apply a Lusternik–Schnirelman pull-tight together with hair-retractions to push any family of tori into Clifford tori, thereby annihilating the relevant relative homotopy groups. A key technical ingredient is the lens-space min-max width, which is realized by a Clifford torus of area $2\pi^2/p$ (index 1), with multiplicity-one ensured in the RP^3 and certain $L(p,q)$ cases by Wang–Zhou. The argument culminates in a parametric retraction to minimal tori, circumventing surgeries required by Ricci-flow approaches and suggesting robustness to higher-genus Heegaard splittings within spherical space-forms.

Abstract

We give a new proof of the Smale conjecture for $\mathbb{RP}^3$ and all lens spaces using minimal surfaces and min-max theory. For $\mathbb{RP}^3$, the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.

The Smale conjecture and min-max theory

TL;DR

This work gives a new, min-max–based proof of the Smale conjecture for lens spaces, including RP^3, by reducing the problem to the homotopy type of spaces of Heegaard tori and then retracting to minimal Clifford tori. The authors classify minimal tori in lens spaces (via Clifford tori), control their parameter spaces with Goeritz groups, and apply a Lusternik–Schnirelman pull-tight together with hair-retractions to push any family of tori into Clifford tori, thereby annihilating the relevant relative homotopy groups. A key technical ingredient is the lens-space min-max width, which is realized by a Clifford torus of area (index 1), with multiplicity-one ensured in the RP^3 and certain cases by Wang–Zhou. The argument culminates in a parametric retraction to minimal tori, circumventing surgeries required by Ricci-flow approaches and suggesting robustness to higher-genus Heegaard splittings within spherical space-forms.

Abstract

We give a new proof of the Smale conjecture for and all lens spaces using minimal surfaces and min-max theory. For , the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.
Paper Structure (14 sections, 29 theorems, 115 equations)

This paper contains 14 sections, 29 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. The Smale conjecture in lens spaces
  3. Space of minimal tori in lens spaces
  4. Goeritz group of lens spaces
  5. Equivalent form of Smale Conjecture in lens spaces
  6. Min-max theorem
  7. Proof of Theorem \ref{['main']}
  8. Lusternik-Schnirelman argument
  9. The case $k=1$
  10. Retracting filigree of tori
  11. Retracting tori in $T^2\times[-r,r]$
  12. Retracting to a single minimal torus
  13. Retraction to a family of minimal tori
  14. Appendix

Key Result

Theorem 1.1

The inclusion $O(4)\to \hbox{Diff}(\mathbb{S}^3)$ is a homotopy equivalence.

Theorems & Definitions (48)

  • Theorem 1.1: Hatcher 1982 H
  • Conjecture 1.2
  • Theorem 1.3: Bamler-Kleiner 2018 BK1,BK3
  • Theorem 1.4: Smale Conjecture for lens spaces
  • Conjecture 1.5
  • Proposition 2.1: Isometry groups of lens spaces
  • Theorem 2.2
  • proof
  • Proposition 2.3: Space of unparameterized minimal tori
  • Proposition 2.4: Goeritz group of lens spaces (Proposition 9.3 JM
  • ...and 38 more