# Isotopy in dimension 4

I will describe why the trivial knot S^2-->S^4 has non-unique spanning discs up to isotopy. This comes from a chain of deductions that include a description of the low-dimensional homotopy-groups of embeddings of S^1 in S^1xS^n (for n>2), a group structure on the isotopy-classes of reducing discs of S^1xD^n, and the action of the diffeomorphism group Diff(S^1xS^n) on the embedding space Emb(S^1, S^1xS^n). Roughly speaking, these results say there is no direct translation from dimension 3 to 4, for the Hatcher-Ivanov theorems on spaces of incompressible surfaces, moreover there is a more direct analogy with the Hatcher-Wagoner results in high dimensions. We have a variety of related results, such as a description of the low-dimensional homotopy groups of the diffeomorphism groups of manifolds such as S^2xD^2 , and the boundary connect-sum of two copies of S^2xD^2 . This is joint work with David Gabai.

Zoom meeting session : https://stanford.zoom.us/j/424763284