Boundary Dehn twists are often commutators

TL;DR

The paper demonstrates that boundary Dehn twists on certain 4-manifolds become trivial in the abelianized smooth mapping class group, providing explicit constructions and generalizing previous results.

Contribution

It explicitly constructs diffeomorphisms whose commutators represent boundary Dehn twists, showing their triviality in the abelianized mapping class group for broad classes of 4-manifolds.

Findings

Boundary Dehn twists are trivial in the abelianized smooth mapping class group for certain 4-manifolds.

Constructed explicit diffeomorphisms representing boundary Dehn twists as commutators.

Generalized previous results on the triviality of boundary Dehn twists on punctured K3 surfaces.

Abstract

For $X$ any complete intersection of even complex dimension or any connected sum thereof (or, more generally, any space among certain broad classes of smooth manifolds), we concretely construct orientation-preserving diffeomorphisms $a,c$ of punctured $X$ rel boundary whose commutator $[a,c]$ represents the smooth mapping class rel boundary of the boundary Dehn twist. This shows that boundary Dehn twists on 4-manifolds known to be nontrivial in the smooth mapping class group rel boundary by work of Baraglia-Konno, Kronheimer-Mrowka, J. Lin, and Tilton become trivial after abelianization, generalizing work of Y. Lin which applied an argument based on the global Torelli theorem and an obstruction of Baraglia-Konno to prove that the abelianized boundary Dehn twist on the punctured $K3$ surface is trivial.