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 spe...

We extend F.Pastjin's construction of uniform decomposable chains of finite rank to those of rank 'finite over a limit' and investigate infinite (length ω) products and unions of such chains. We derive an extension of Pastjin's characterisation to uniform decomposable chains of small transfinite rank (≤ ω + ω).

Working in the context of reverse mathematics, we give a fine-grained characterization result on strength two possible definitions for Effective Transfinite Recursion used literature. Moreover, show that Π20-induction along well-order X is equivalent to statement exponentiation any power well-founded.