On Torelli groups and Dehn twists of smooth 4-manifolds

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 $\mathrm{\partial }X\cong S^3$ is trivial after taking connected sums with enough copies of $S^2\times S^2$.