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On the Smale Conjecture for Diff$(S^4)$

Selman Akbulut

Abstract

Recently Watanabe disproved the Smale Conjecture for $S^4$, by showing Diff$(S^{4})\neq SO(5)$. He showed this by proving that their higher homotopy groups are different. Here we prove this more directly by showing $π_{0}$Diff$(S^{4})\neq 0$, otherwise a certain loose-cork could not possibly be a loose-cork.

On the Smale Conjecture for Diff$(S^4)$

Abstract

Recently Watanabe disproved the Smale Conjecture for , by showing Diff. He showed this by proving that their higher homotopy groups are different. Here we prove this more directly by showing Diff, otherwise a certain loose-cork could not possibly be a loose-cork.

Paper Structure

This paper contains 3 sections, 1 theorem, 5 equations, 14 figures.

Table of Contents

  1. An exotic diffeomorphism
  2. Forming the infinite order cork
  3. Constructions and proofs

Key Result

Theorem 1

The diffeomorphisms $\phi_{n}^{0}:(B^4, S^3)\to (B^4, S^3)$ are not isotopic to each other fixing the boundary, for distinct integers $n>0$.

Figures (14)

  • Figure 1: Dehn twist
  • Figure 2: $\phi_{n}^{0} : B^{3}\times J \to B^{3}\times J$
  • Figure 3: $\phi^{o}_{n} : B^{3}\times J \to B^{3}\times J$
  • Figure 4: Forming $D_{n}$ by swallow-follow isotopies
  • Figure 5: Swallow-follow isotopies of $K\#-K$
  • ...and 9 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Remark 2