Transfinite Milnor invariants for 3-manifolds

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