On Topologically Unknotted Spheres

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1. B. Mazur [10] and M. Brown [2] have contributed to a general theorem on the unknotting of (n - 1)-spheres in the n-sphere. In its most recent form [4], it states that if X is a subset of the n-sphere Sn, where X is homeomorphic to Sn-1, and if X satisfies a topological condition of local smoothness, then there is a homeomorphism of Sn onto itself which takes X onto an equatorial (n - 1)-sphere of Sn. One wonders whether an analogous statement is true about subsets Y of Sn, where Y is homeomorphic to Sk, for values of k other than k = n - 1. It is a classical fact that knotted smooth spheres occur when k = n - 2. But otherwise there are no known examples of truly knotted spheres Sk in Sn, except when there is some sort of local pathology. Here we shall show that a locally smooth k-sphere embedded in So is indeed topologically unknotted, provided k 5. The case of a locally smooth 1-sphere in S4 is unsolved. A condition will be given which insures that an (n - 2)-sphere X contained smoothly in Sn is unknotted. It is that n > 5 and Sn - X have the homotopy type of S1. The case of a 2-sphere in S4 is unsolved. As for 1-spheres in S3, the problem has been settled by a special result in 3-dimensional topology [1] and by Dehn's lemma [11]. When the k-sphere X contained in Sn is smooth except possibly at one point, and k 5, then it is shown that X is unknotted. The case of knots which may fail to be smooth at two points cannot be handled by this method. This odd state of affairs appears related to the difficulties in the isotopy conjecture, that a homeomorphism of degree one of Sn on itself should be isotopic to the identity map. The proof of these results is rather complicated. It is necessary to study in detail homotopy properties of the complement of a smooth knot. A delicate application of the engulfing theorem [12] is made. The proof is completed by applying a result about the union of open cones [13]. Comparion with the piecewise-linear unknotting theorem of E. C. Zeeman [15] shows this. Zeeman's theorem, and ours, have an analogous appearance and a region of overlap. Furthermore, both are proved by somewhat similar piecewise-linear methods. Zeeman's result is much stronger than ours when applied to piecewise-linear embeddings of