Higher dimensional cable knots and their finite cyclic covering spaces

Abstract

As an analogue of the classical cable knot, the p-cable n-knot about an n-knot K, where p is an integer and n⩾2, is defined, and some basic properties of higher dimensional cable knots are described. We show that for p>0 then p-fold branched cyclic covering space of an ( n+2)-sphere branched over the p-cable knot about an n-knot K is an ( n+2)-sphere or a homotopy ( n+2)-sphere 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, we prove that for any n⩾2 and p⩾2, the composition of p copies of any n-knot K is the fixed point set of a Z p -action on an ( n+2)-sphere. This is another counterexample to the higher dimensional Smith conjecture.