On Torelli groups and Dehn twists of smooth 4-manifolds

TL;DR

This paper explores the structure of Torelli groups and Dehn twists in smooth 4-manifolds, extending recent results and providing new proofs for the triviality of certain boundary Dehn twists after stabilization.

Contribution

It generalizes Gay's recent findings on the smooth mapping class group of $S^4$ and offers an alternative proof regarding the triviality of boundary Dehn twists after connected sums.

Findings

Generalized Gay's result on smooth mapping class groups of $S^4$

Proved boundary Dehn twists become trivial after enough connected sums with $S^2\times S^2$

Provided an alternative proof for Saeki's consequence on Dehn twists

Abstract

This note has two related but independent parts. Firstly, we prove a generalisation of a recent result of Gay on the smooth mapping class group of $S^4$. Secondly, we give an alternative proof of a consequence of work of Saeki, namely that the Dehn twist along the boundary sphere of a simply-connected closed smooth $4$-manifold $X$ with $\partial X\cong S^3$ is trivial after taking connected sums with enough copies of $S^2\times S^2$.