Family Bauer--Furuta invariant, Exotic Surfaces and Smale conjecture

Abstract

We establish the existence of a pair of exotic surfaces in a punctured K3 which remains exotic after one external stabilization and have diffeomorphic complements. A key ingredient in the proof is a vanishing theorem of the family Bauer-Furuta invariant for diffeomorphisms on a large family of spin 4-manifolds, which is proved using the Tom Dieck splitting theorem in equivariant stable homotopy theory. In particular, we prove that the S 1 -equivariant family Bauer-Furuta invariant of any orientation-preserving diffeomorphism on S 4 is trivial and that the Pin(2)equivariant family Bauer-Furuta invariant for a diffeomorphism on S 2 × S 2 is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic selfdiffeomorphisms on S 4 or S 2 ×S 2 . En route, we observe a curious element in the Pin(2)-equivariant stable homotopy group of spheres which could potentially be used to detect an exotic diffeomorphism on S 4 .

Stabilization distance bounds from link Floer homology

“…It is an open conjecture whether a single copy of 𝕊 2 × 𝕊 2 always suffices. Recently, Lin and Mukherjee [25] constructed a pair of surfaces-with-boundary in a punctured K3 that are topologically isotopic but not smoothly isotopic, and remain so after stabilizing their complements once with 𝕊 2 × 𝕊 2 . They showed this using family Bauer-Furuta invariants.…”

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

“…Recently, Baraglia [8] gave exotic surfaces in closed 4-manifolds whose complements are diffeomorphic, based on a method closely related to our technique. Also, in [45], J. Lin and the third author gave exotic surfaces in the punctured K3 whose complements are diffeomorphic using the 4-dimensional Dehn-twist and the familes Bauer-Furuta invariant.…”