The piecewise linear unknotting of cones

A POLYHEDRON X is said to unknot in another polyhedron Y, if any two piecewise linear embeddings of X and Y that are homotopic are also ambient isotopic; this means that one embedding can be carried to the other by a piecewise linear isotopy of Y. If Y happens to be a polyhedral sphere or ball, this definition can be simplified by reference to the result of Alexander and Gugenheim [3], that any piecewise linear orientation preserving homeomorphism of Y is, in this case, isotopic to the identity. The theory of unknotting when X and Y are both manifolds has recently been much studied, taking its impetus from the theorems of Zeeman [9]. These show that Sp, the p-dimensional sphere, unknots in S”, and that proper embeddings (i.e. with boundary embedded in boundary) of the p-ball BP, in B”, also unknot, provided that in both cases, (n -p) 2 3. The apparatus developed by Zeeman for the proof of these results is here examined and extended. The main theorem (Theorem (1)) obtained in this paper is that if (B”, Xp) is a pair consisting of a ball B", with a cone X embedded in it, with the base of the cone, and only the base of the cone, embedded in the boundary of the ball, then (B", Xp) is pairwise homeomorphic to the cone on (as", dB"n P), by a homeomorphism that is the identity on as", this being subject to the usual proviso that (n p) 2 3. This result leads immediately to theorem 5 which is a suspension theorem for the property of “being unknotted in a sphere”. More precisely it is shown that if. Xp unknots in S”, (n p) 2 3, then the r-fold suspension of X unknots in Sn+r.