On absolutely exotic diffeomorphisms of 4-manifolds
Hokuto Konno, Abhishek Mallick, Masaki Taniguchi
TL;DR
The paper addresses the problem of producing absolutely exotic diffeomorphisms on contractible 4-manifolds, i.e., diffeomorphisms nontrivial in the diffeomorphism mapping class group yet trivial on the boundary homeomorphism level. It develops a method to upgrade relatively exotic diffeomorphisms to absolute ones using Akbulut--Ruberman invertible homology cobordisms, yielding infinitely many pairwise non-diffeomorphic contractible 4-manifolds with absolutely exotic diffeomorphisms of infinite order. The boundaries of these manifolds are Haken and their boundary Heegaard Floer invariants $HF^{red}$ grow, enabling a distinction among the examples. The work extends the phenomenon to higher homotopy groups of the diffeomorphism group, showing that relative exotica in $\pi_n(Diff_{\partial}(W))$ give absolutely exotic elements in $\pi_n(Diff(V))$, and connects to known higher-order exotica via Watanabe’s and Auckly–Ruberman constructions. Overall, it broadens the landscape of exotic diffeomorphisms in dimension four and links 4-manifold topology with 3-manifold invariants through cobordisms and Floer theory.
Abstract
We prove that there exist infinitely many contractible compact smooth $4$-manifolds $C$ that admit absolutely exotic diffeomorphisms of infinite order in $π_0(\mathrm{Diff}(C))$. By ``absolutely", we mean that isotopies are not required to be relative to the boundary. This follows from a theorem that produces absolutely exotic diffeomorphisms from relatively exotic diffeomorphisms, analogous to a theorem of Akbulut and Ruberman that produces absolutely exotic 4-manifolds from relatively exotic 4-manifolds.