Unknotting 3-spheres in six dimensions
E. Christopher Zeeman · Proceedings of the American Mathematical Society · 1962
Haefliger [2] has shown that a differentiable embedding of the 3-sphere S3 in euclidean 6-dimensions El can be differentiably knotted.On the other hand any piecewise linear embedding of Sn in Ek is combinatorially unknotted if k^n + 3 (see [5; 6; 7]).The case S3 in E6 appears to be the first occasion on which the differentiable and combinatorial theories of isotopy differ.Therefore it seemed worthwhile to give separately the proof of the combinatorial unknotting in S3 in E6, because the argument in this case is considerably simpler than that in the general case [7], The proof is similar to that of unknotting S2 in E6 (see [5]), although it does involve one new idea, that of "severing the connectivity of the near and far sets" (without which I had conjectured the opposite in Remark 2 of [S]). Theorem.Given a piecewise linear embedding of a combinatorial 3-sphere S is euclidean 6-dimensions E*, then it is unknotted, i.e. there is a piecewise linear homeomorphism of 7s8 onto itself, throwing S onto the boundary of a 4-simplex.