#### Volume 3, issue 1 (2003)

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On 4–fold covering moves

### Nikos Apostolakis

Algebraic & Geometric Topology 3 (2003) 117–145
 arXiv: math.GT/0302225
##### 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 ${S}^{3}$.

##### Keywords
branched covering, covering move, colored braid, colored link, $3$–manifold
Primary: 57M12
Secondary: 57M25