On the mapping class group of 4-dimensional 1-handlebodies via Budney-Gabai invariants

We define an invariant $(W_3)_m$ for $\pi_0\mathrm{Diff}(\natural_m S^1\times D^3,\partial)$ for $m\geq 1$ that generalizes Budney--Gabai's $W_3$ invariant. We give a computational framework inspired by Budney--Gabai and use it to calculate the invariant for all unknotted barbell difeomorphisms of $\natural_m S^1\times D^3$ for $m=1,2$. This allows us to detect more linearly independent elements in $\pi_0\mathrm{Diff}(S^1\times D^3,\partial)$, and to prove that $\pi_0\mathrm{Diff}( \natural_2 S^1\times D^3,\partial)/ \left( \pi_0 \mathrm{Diff}(S^1\times D^3,\partial)\right)^2$ admits infinitely generated subgroups generated by unknotted barbell diffeomorphisms, leading to infinitely many properly embedded separating 3-balls that are non-isotopic relative to the boundary.