On the mapping class groups of 4-manifolds with 1-handles

We develop a framework that generalizes Budney-Gabai’s $W_3$ invariant on ${\pi }_0\text{Diff}(S^1\times D^3,\mathrm{\partial })$ to 4-manifolds with 1-handles. As applications, we show that if $M=(S^1\times D^3)\mathrm{♮}\widehat{M}$ where $\widehat{M}$ either has the form $I\times Y$ or is a punctured aspherical manifold, then the center of the mapping class group of $M$ is of infinite rank.