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

TL;DR

This paper constructs exotic surfaces in a punctured K3 surface, proves a vanishing theorem for family Bauer--Furuta invariants on certain 4-manifolds, and shows these invariants cannot detect some exotic diffeomorphisms.

Contribution

It introduces a vanishing theorem for the family Bauer--Furuta invariant on large classes of spin 4-manifolds, impacting the detection of exotic diffeomorphisms.

Findings

Existence of exotic surfaces in punctured K3 surfaces.

Vanishing of family Bauer--Furuta invariants for certain diffeomorphisms.

Invariants do not detect exotic self-diffeomorphisms on S^4 or S^2×S^2.

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 $\mathrm{Pin}(2)$-equivariant family Bauer--Furuta invariant for a diffeomorphism on $S^{2}\times S^{2}$ is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic self-diffeomorphisms on $S^{4}$ or $S^{2}\times S^{2}$. Furthermore, our theorem also applies to certain exotic loops of diffeomorphisms on $S^{4}$ (as recently discovered by Watanabe) and show that these loops have trivial family Bauer--Furuta invariants. En route, we observe a curious element in the $\mathrm{Pin}(2)$-equivariant stable homotopy group of spheres which could potentially be used to detect an exotic diffeomorphism on $S^{4}$.