  We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere. For
wave maps of topological degree zero we prove global existence and scattering
for energies below twice the energy of harmonic map, Q, given by stereographic
projection. We deduce this result via the concentration compactness/rigidity
method developed by the second author and Merle. In particular, we establish a
classification of equivariant wave maps with trajectories that are pre-compact
in the energy space up to the scaling symmetry of the equation. Indeed, a wave
map of this type can only be either 0 or Q up to a rescaling. This gives a
proof in the equivariant case of a refined version of the threshold conjecture
adapted to the degree zero theory where the true threshold is 2E(Q), not E(Q).
The aforementioned global existence and scattering statement can also be
deduced by considering the work of Sterbenz and Tataru in the equivariant
setting.
  For wave maps of topological degree one, we establish a classification of
solutions blowing up in finite time with energies less than three times the
energy of Q. Under this restriction on the energy, we show that a blow-up
solution of degree one is essentially the sum of a rescaled Q plus a remainder
term of topological degree zero of energy less than twice the energy of Q. This
result reveals the universal character of the known blow-up constructions for
degree one, 1-equivariant wave maps of Krieger, the fourth author, and Tataru
as well as Raphael and Rodnianski.
