We prove the existence of a finite set of moves sufficient to relate any two representations of the same –manifold as a –fold simple branched covering of . 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 and can be viewed as a second step in a program to establish similar results for arbitrary degree coverings of .
"On 4–fold covering moves." Algebr. Geom. Topol. 3 (1) 117 - 145, 2003. https://doi.org/10.2140/agt.2003.3.117