Exoticity of the commutator diffeomorphism α^0 in the stabilized setting

Prove that, in the stabilized setting where X and X′ are distinct smooth structures on the same underlying closed, oriented 4-manifold, U is a stabilizing manifold, φ: X # U → X′ # U is a diffeomorphism inducing the same homology isomorphism as ψ # 1_U for some homeomorphism ψ: X → X′ and preserving the embeddings of the specified surface Σ, and f: U → U is a diffeomorphism acting non-trivially on H_2(U), the commutator α^0 = φ ∘ (1_{X′} ∘ f) ∘ φ^{-1} ∘ (1_X ∘ f)^{-1}: Z_0 → Z_0 (with Z_0 = X # U) is an exotic diffeomorphism of Z_0 (i.e., smoothly non-isotopic to the identity).

Background

The paper develops a recursive strategy for constructing exotic diffeomorphisms and families thereof by stabilizing pairs of homeomorphic but non-diffeomorphic 4-manifolds. In this framework, one starts with distinct smooth structures X and X′ that become diffeomorphic after connected sum with a simple stabilizing manifold U, together with a diffeomorphism φ: X # U → X′ # U compatible with a chosen embedded surface Σ.

A key ingredient is a non-trivial diffeomorphism f of U, typically built from reflections in embedded spheres. The commutator construction α^0 = [φ, 1 ∘ f] is suggested as the source of exotic diffeomorphisms, mirroring earlier examples where such commutators were shown to be nontrivial via gauge-theoretic invariants. The authors note that the original construction in Ruberman (1998) fits this pattern and that additional instances have been found, but a general proof covering the full stabilized setting remains conjectural.