# Isotoping Watanabe's theta graph diffeomorphism of S^4

# Isotoping Watanabe's theta graph diffeomorphism of S^4

Short version: I will draw pictures of diffeomorphisms of S^4 and of isotopies between diffeomorphisms of S^4. Long version: Watanabe disproved the smooth 4-dimensional Smale conjecture using a construction which turns trivalent graphs in 4-manifolds into parameterized families of diffeomorphisms of 4-manifolds. This exhibited nontrivial homotopy groups of Diff(B^4,S^3), but did not say anything about pi_0. However, the construction does give a diffeomorphism of B^4 (or equivalently of S^4) that is at least a candidate for a nontrivial element of pi_0(Diff^+(S^4)), starting with the simplest interesting trivalent graph, the theta graph. In earlier work with Daniel Hartman I studied the "(1,2)-subgroup" of pi_0(Diff^+(S^4)), represented by diffeomorphisms that can be studied Cerf theoretically described using only handles of index 1 and 2, and we showed that this subgroup has at most 2 elements. In this talk I will outline the proof that Watanabe's theta graph diffeomorphism is isotopic to the identity if and only if this (1,2)-subgroup is trivial. Part of the point is to illustrate some explicit techniques for demonstrating relations in the smooth mapping class groups of 4-manifolds, and part of the point is to exhibit some interesting examples.