The Generalized Smale Conjecture for 3-manifolds with genus 2 one-sided Heegaard splittings

TL;DR

This paper proves the Generalized Smale Conjecture for most 3-manifolds with genus 2 one-sided Heegaard splittings, extending previous results to new cases except for a specific lens space.

Contribution

It extends the proof of the Generalized Smale Conjecture to all remaining cases with genus 2 one-sided Heegaard splittings, excluding L(4,1).

Findings

Proves the conjecture for all remaining cases with one-sided Klein bottles.

Completes the classification for manifolds with genus 2 one-sided splittings.

Identifies the exception of lens space L(4,1).

Abstract

The Generalized Smale Conjecture asserts that if M is a closed 3-manifold with constant positive curvature, then the inclusion of the group of isometries into the group of diffeomorphisms is a homotopy equivalence. For the 3-sphere, this was the classical Smale Conjecture proved by A. Hatcher. N. Ivanov proved the Generalized Smale Conjecture for the M which contain a 1-sided Klein bottle and such that no Seifert fibering is nonsingular on the complement of any vertical Klein bottle. We prove it in all remaining cases containing a one-sided Klein bottle, except for the lens space L(4,1).