  The configuration space of the planar three-body problem when collisions are
excluded has a rich topology which supports a large set of free homotopy
classes. Most classes survive modding out by rotations. Those that survive are
called the reduced free homotopy classes and have a simple description when
projected onto the shape sphere. They are coded by syzygy sequences. We prove
that every reduced free homotopy class, and thus every reduced syzygy sequence,
is realized by a reduced periodic solution to the Newtonian planar three-body
problem. The realizing solutions have nonzero angular momentum, repeatedly come
very close to triple collision, and have lots of "stutters"--repeated syzygies
of the same type. The heart of the proof is contained in the work by one of us
on symbolic dynamics arising out of the central configurations after the triple
collision is blown up using McGehee's method.
