  We construct the first known examples of nontrivial, normal, all
pseudo-Anosov subgroups of mapping class groups of surfaces. Specifically, we
construct such subgroups for the closed genus two surface and for the sphere
with five or more punctures. Using the branched covering of the genus two
surface over the sphere and results of Birman and Hilden, we prove that a
reducible mapping class of the genus two surface projects to a reducible
mapping class on the sphere with six punctures. The construction introduces
"Brunnian" mapping classes of the sphere, which are analogous to Brunnian
links.
