Incomplete 02/29/08: Lifting Invariants for Nielsen classes with <i>G</i>=S<sub>n</sub>, with emphasis on using the Schur multiplier to figure braid orbits.  
<span style="color: rgb(153, 51, 153);">When G=S</span><sub
 style="color: rgb(153, 51, 153);">n</sub><span
 style="color: rgb(153, 51, 153);">, </span><span
 style="font-weight: bold; color: rgb(153, 51, 153);">C' </span><span
 style="color: rgb(153, 51, 153);">has one class, that of a 2-cycle:</span>
Then the indexing set is just the number of 2-cycles, <span
 style="font-style: italic;">r</span> &#8805; 2(<span
 style="font-style: italic;">n</span> -1) (Riemann-Hurwitz). Then, it
is the oldest argument in the book that
there is one orbit if this condition holds. Voelklein's book, Lem. 10.15
book repeats my (essentially) 2-line argument. Since the Schur multiplier of
<span style="font-style: italic;">S<sub>n</sub></span> has order 4, it <i>is</i> surprising that this Schur multiplier does not split the Nielsen class into braid orbits.  We also explain why not.  <br>