Knotted 3-balls in S^4
Abstract
The unknot U in S^4 has non-unique smooth spanning 3-balls up to isotopy fixing U. Equivalently there are properly embedded non-separating 3-balls in S^1xB^3 not properly isotopic to 1xB^3. More generally there exist non-separating 3-spheres in S^1xS^3 not isotopic to 1xS^3 and non trivial elements of \pi_0 Diff_0(S^1xS^3). Along the way we introduce barbell diffeomorphisms, implantations and twistings to construct and modify diffeomorphisms homotopic to the identity. We also introduce a 2-parameter calculus of embeddings of the interval into 4-manifolds and introduce a framed cobordism method as well as a direct method for showing that certain 2-parameter families are homotopically non trivial and diffeomorphisms are isotopically nontrivial. Extensions to higher dimensional manifolds are obtained.
Cite
@article{arxiv.1912.09029,
title = {Knotted 3-balls in S^4},
author = {Ryan Budney and David Gabai},
journal= {arXiv preprint arXiv:1912.09029},
year = {2021}
}
Comments
98 pages, 66 figures. v3: Replaced Theorem 3.1 from v2 whose proof had an error. The new proofs use the scanning construction. There is also the new conclusion that \pi_0 Diff(S^1 x S^3) is not finitely generated