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

Abstract: We define an invariant $(W_3)_m$ for $$π_0\mathrm{Diff}(\natural_m S^1\times D<sup>3,\partial)$$ 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 $\natural_m S^1\times D<sup>3$ for $m=1,2$. This allows us to detect more linearly independent elements in $$π_0\mathrm{Diff}(S^1\times D<sup>3,\partial)$$, and to prove that $$π_0\mathrm{Diff}( \natural_2 S^1\times D^3,\partial)/ \left( π_0 \mathrm{Diff}(S^1\times D<sup>3,\partial)\right)<sup>2$$ 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.