C(onway)F(ried)P(arker)V(oelklein) connectedness results


There are at least five connectedness results – three #1, #4 and #5 should now be considered classical – that give precise knowlege on Hurwitz space components (equivalently, braid orbits on Nielsen classes).
  1. Moduli space of curves of genus g: Connectedness of Hurwitz spaces defined by 2-cycles in Sn (Clebsch).
  2. Hurwitz spaces defined by 3-cycles and the parity of a particular linear system (Fried).
  3. Spaces of genus 0 pure-cycle covers (Liu-Osserman).
  4. Spaces of elliptic curves with n-torsion points – modular curves (Who knows who first observed this?).
  5. Hurwitz spaces for Nielsen classes including all conjugacy classes in the defining group sufficiently often (Conway-Fried-Parker-Voelklein).
We explain these, then give a much improved version of #5. Unlike the present #5, this version has significant overlap with #1, #2 and #4. Better yet it points to an umbrella result over both #2 and #3.

All these connectedness results show the significance of identifying components by their cusps. We also see the natural conditions – Fried-Serre-Weigel – for forming Modular Towers over a given a Hurwitz space.

I. 3-cycle Hurwitz spaces and Spin Invariants
  1. Quick start on Schur Multipliers
  2. R/G Lifting Invariant if C is |R/G|'
  3. A Formula for the Spin-Lift Invariant
  4. Main Theorem: Hr orbits on Ni(An,C3r), r ≥ n-1 ≥ 3
  5. Constellation of spaces H(An,C3r)abs
  6. Braiding 3-cycle Nielsen classes to Normal Form
  7. Does intricate finite group theory make understanding Hurwitz spaces hopeless?

II. Lessons from Alternating Group Hurwitz spaces
  1. Dropping the |R/G|' condition
  2. Combine 2-cycle and 3-cycle cases for CFPV
  3. Definition and advantage of g-p' reps.
  4. Find a co-final set I of G, so each G∈ I has ∞-ly many Q Hurwitz space components [FrV91, Prop. 1]
  5. For (G,p) (p-perfect G), find when ∃ ∞-ly many MTs with all levels with Q components [Fr95, Thm. 3.21]
Appendices:
  1. Liu-Osserman pure-cycle cases with unbraidable outer automorphisms
  2. Higher Order g-p' representatives
  3. Comments on Schur multiplier of Sn
  4. Using the sh-incidence Matrix for (A4, C±32)