Hughes, Kim and I recently showed that for any n>1, there exists a pair of 3-dimensional genus-n solids in the 4-sphere with the same boundary, and that are homeomorphic relative to their boundary, but that do not become isotopic rel boundary even when their interiors are pushed into the 5-dimensional ball. This proves a conjecture of Budney and Gabai (who previously constructed 3-balls in the 4-sphere with the same boundary that are not isotopic rel boundary) for n>1 in a very strong sense, and also provides as a new example of "similar" objects remaining "different" when stabilized or when given an extra dimension of freedom -- in contrast, it is unknown whether two genus-g Seifert surfaces in S^3 with the same boundary must become isotopic rel. boundary when pushed into B^4 (and this is the first analogous counterexample to that in B^5).
In this talk, I’ll give 3D and 4D motivation for this result, state some cool theorems about surfaces in S^4, and sketch how to construct interesting codimension-2 knotting in dimensions 3, 4, and 5.
This is joint work with Mark Hughes (BYU) and Seungwon Kim (IBS-CGP)