  In his 1957 paper, John Milnor introduced link invariants which measure the
homotopy class of the longitudes of a link relative to the lower central series
of the link group. Consequently, these invariants determine the lower central
series quotients of the link group. This work has driven decades of research
with profound influence. One of Milnor's original problems remained unsolved:
to extract similar invariants from the transfinite lower central series of the
link group. We reformulate and extend Milnor's invariants in the broader
setting of 3-manifolds, with his original invariants as special cases. We
present a solution to Milnor's problem for general 3-manifold groups,
developing a theory of transfinite invariants and realizing nontrivial values.
