On 4–fold covering moves

Abstract

We prove the existence of a finite set of moves sufficient to relate any two representations of the same $3$–manifold as a $4$–fold simple branched covering of $S^3$. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degree $3$ and can be viewed as a second step in a program to establish similar results for arbitrary degree coverings of $S^3$.