Our primary aim is to show that if \( \widetilde{\Sigma} \) is the \( n \)-fold cyclic branched covering, \( n > 1 \), of a prime knot \( K \) in a homotopy 3-sphere \( \Sigma \), such that \( \Sigma - K \) is irreducible and contains a nonperipheral incompressible surface, then \( \widetilde{\Sigma} \) is not a homotopy sphere. This will be achieved by using the equivariant loop theorem to show that \( \widetilde{\Sigma} \) contains an incompressible surface of positive genus. Actually, we shall work in the more general context of a regular branched covering of a link in an arbitrary closed, orientable 3-manifold, as this is needed for the study of noncyclic finite group actions on homotopy 3-spheres. Our main result is Theorem 1, which asserts that, under certain circumstances, an incompressible surface in the complement of a link gives rise to one in any regular branched covering of the link. The special case of incompressible tori is considered in Theorem 2; here the proof actually uses the Smith conjecture. Interpreting Theorem 2 in terms of hyperbolic structures, using Thurston’s uniformization theorem for Haken manifolds (see Thurston [1979] and Chapter V, this volume by Morgan) we obtain Corollary 2.1, which states that if a regular branched covering of a link is hyperbolic, then so is the complement of the link.
These results are stated in Section 2, which also contains the necessary definitions and terminology. The proofs of Theorems 1 and 2 and Corollary 2.1 are given in Section 3. Finally, in Section 4, we give an elementary proof of the equivariant loop theorem for involutions (Theorem 3). A similar proof has been given by Kim and Tollefson [1980].
