Knotted handlebodies in the 4-sphere and 5-ball

Abstract

For every integer g ≥ 2 we construct smooth 3-dimensional 1handlebodies H1 and H2 of genus g, which are properly embedded in B 5 with the same boundary, such that both H1 and H2 are smoothly boundary parallel and are homeomorphic rel boundary as 3-manifolds, yet H1 and H2 are not related by a topological, locally flat isotopy rel boundary.In other words, we construct 3-dimensional genus-g 1-handlebodies smoothly embedded in S 4 with the same boundary that are defined by the same cut systems of their boundary yet are not isotopic rel boundary via any locally flat isotopy even when their interiors are pushed into B 5 . This in particular proves a conjecture of Budney-Gabai for g ≥ 2.

Unknotting 3-balls in the 5-ball

“…We will work mostly in the smooth category, with a slight detour into the PL-category. The main argument for unknotting any B 3 is derived from the fact that any nonseparating S 3 in S 1 × S 3 extends to a smoothly embedded B 4 in S 1 × B 4 . This is a consequence of the classification of pseudoisotopies of S 1 × S 3 ( [5]).…”

“…As we are considering what happens in B 5 , take a collar neighborhood S 4 × I of the boundary and extend Y to Y× I. As U is the unknot, Y is diffeomorphic to S 1 × B 3 and Y × I is then diffeomorphic to S 1 × B4 , which in our given product structure is just (S 1 × B 3 ) × I. Now consider the 3-spheres, B × {0} ∪ U × I ∪ B × {1}.…”