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

Abstract

Recently Watanabe disproved the Smale Conjecture for $S^4$, by showing Diff$(S^4)\ne SO(5)$. He showed this by proving that their higher homotopy groups are different. Here we prove this more directly by showing $\ddot{I}{€}_0$Diff$(S^4)\ne 0$, otherwise a certain loose-cork could not possibly be a loose-cork.

9 pages, 14 figures, expanded version