Higher dimensional cable knots and their finite cyclic covering spaces (Q1063908)

For an n-knot K in \(S^{n+2}\), \(n\geq 2\), we define a p-cable n-knot about K, which we denote by \(\Gamma\) (K;p), as follows: Let U be a trivial n-knot and N(U) its tubular neighborhood. Let \(\ell\) be a simple closed curve in \(S^{n+2}-N(U)\) representing \(x^ p\), where \(x\in \pi_ 1(S^{n+2}-U)\) is a meridian of U. Let V be a tubular neighborhood of \(\ell\) in \(S^{n+2}-N(U)\). Let \(h: S^{n+2}-int V\to N(K)\) be a homeomorphism. We define \(\Gamma (K;p)=h(U)\). It is proved that for \(p>0\) the p-fold branched cyclic covering space of \(S^{n+2}\) branched over \(\Gamma\) (K;p) is \(S^{n+2}\) or a homotopy \(S^{n+2}\) which is the result of Gluck-surgery on the composition of p copies of K according as if p is odd or even. At the same time, it is proved that for any \(n\geq 2\) and \(p\geq 2\), the composition of p copies of any n-knot K is the fixed point set of a \({\mathbb{Z}}_ p\)-action on \(S^{n+2}\). This is another counterexample to the higher dimensional Smith conjecture.