3-fold branched coverings and the mapping class group of a surface

On 4–fold covering moves

“…This is done in Section 3 which culminates in Theorem 15, result that which gives normal generators of the kernel of the lifting homomorphism. The proof of this theorem is very similar to the proof of the corresponding theorem in [4].…”

“…The seemingly intractable calculations are made possible by the following two observations. First, the existence of the exceptional homomorphism S 4 S 3 , which we call "dimming the lights," means that one only needs to study the action of the 3-fold stabilizer already computed in [4] instead of the action of the full braid group. Second, the two "local moves," can already be considered at this level as moves between braids and therefore one only needs to compute in the "reduced" picture i.e.…”

“…An excellent background source is [2] and the references contained therein. We also assume the reader is familiar with the contents of [4] as well as with Piergallini's papers regarding covering moves [8], [9]. Finally we mention that the author's thesis [1] contains extensive background material.…”

Covering moves

“…§0), and the homeomorphism a2yxaxy0a{yxa2, given by y2, makes b2 into d2 and vice versa. In order to prove the second fact, let us consider the characterization of kerA (given in [2], where the homomorphism X is denoted by I) as the smallest normal subgroup of L(2g + 4) containing the following elements: Then, we are reduced to showing how to insert such four braids in a normalized diagram (that is something more than we really need, so this is a point where our work could be possibly improved).…”

“…[1]), in such a way that we can apply results of [10] and [2], respectively in order to realize stable equivalence of splittings and to relate different braids representing the same splitting homeomorphism. An application of our moves will be given in a forthcoming paper, in which we prove that any 4-manifold can be represented as a 4-fold branched covering of S\…”