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

Abstract

We define an invariant $(W_3{)}_m$ for ${\pi }_0\mathrm{Diff}({♮}_m{S}^1\times D^3,\partial )$ for $m\ge 1$ 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 ${♮}_m{S}^1\times D^3$ 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 ${\pi }_0\mathrm{Diff}({♮}_2{S}^1\times D^3,\partial )/{\left({\pi }_0\mathrm{Diff}(S^1\times D^3,\partial )\right)}^2$ 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.

arXiv comment44 pages, 46 figures