III. Families of covers and the introduction of braids:

Hurwitz spaces are moduli spaces of covers of P1 with �xed monodromy group1 G (given as a subgroup of the symmetric group Sd with d the degree of the cover) and with a �xed number r ≥ 3 of branch points. The basic notation for it is Hr,G and a point representing a cover f, or more exactly its equivalence class, is denoted by [f]. There are several variants of Hurwitz spaces, depending �rst, on whether one is interested in -the mere cover situation: the covers are not necessarily Galois, or -the G-cover situation: the covers are Galois covers given with an isomorphism between their automorphism group and the group G, and, second, on which cover equivalence is used: -the original equivalence: two covers f : X P1 and g : Y P1 are equivalent if there →→ exists an isomorphism χ : XY such that gχ = f, or →◦ -the PGL2-reduced equivalence: f : X P1 and g : Y P1 are equivalent if there exist →→ two isomorphisms χ : XY and α : P1 P1 such that gχ = αf, →→ ◦◦ equivalently, the monodromy group is the Galois group of the Galois closure of the cover. with for both equivalences the extra condition that χ : XY be compatible with the → actions of G in the G-cover situation. For simplicity, we will not distinguish here between these different variants.

IV. Interpreting Hurwitz space components as braid orbits on Nielsen classes:

 By component we mean irreducible or connected components; due to smoothness of H∞, these coincide. r,G 3  There is a classical outer action of the → Hurwitz braid group6 Hr = πtop(Ur, t) on πtop(P1(C) \ t,t0), which induces an action 11 on the �ber Ψ−r 1 via maps BCDΓ. r (t), and on ni(C)• This induced action on Ψ−1(t) is the monodromy action corresponding to the topological cover Ψr : H∞(C) →Ur(C). It r,G can be explicitly determined: π1(Ur, t)top has generators Q1,...,Qr−1 whose action on Ψ−r 1(t), when computed relative to some suitable topological bouquet Γ, corresponds to the following action on ni(C)•: (g1,...,gr) Qi (g1,...,gi−1,gigi+1g−1 ,gi,gi+2,...,gr),i =1,...,r − 1. i −−−→ Components of H∞(C) correspond to orbits of the Hurwitz braid group action. r,G i.e., a r-tuple Γ=(Γ1,...,Γr) of homotopy classes of sample loops based at some point to∈/t generating the topological fundamental group πtop(P1(C)\t,t0) with the unique relation Γ1···=1 (plus some other technical conditions). 1 Γr where BCD stands for “branch cycle description�. The Hurwitz braid group Hr has a classical presentation: it is the group on r−1 generators Q1,...,Qr−1 with relations QiQj =Qj Qi for |i−j|>1, Qi+1QiQi+1=QiQi+1Qi for 1≤i≤r−2 and Q1···Q1=1. Qr−1Qr−1···

V. Examples from dihedral groups and pure-cycles: At this time applications where G=An are still very significant. So, we use pure-cycle Nielsen classes where An arises often, to simultaneously address examples and serious applications without difficult group theory. For example, the Nielsen class Ni(An,C12C1) is non-empty,

Do the three 5-cycle example from I.

VI. Why do we need (the complication of) Nielsen classes?:


[#2] M. Fried, Fields of definition of function fields and Hurwitz families and groups as Galois groups, Communications in Algebra 5 (1977), 17–82.