Transfinite Milnor invariants for 3-manifolds (Q6566458)

For links in a \(3\)-manifold \(M\), Milnor introduced in 1957 invariants obtained from the lower central series of the link group \(\Gamma\) and posed the problem to extract information from the \textit{transfinite} lower series quotients of \(\Gamma\), indexed by ordinals. In the present paper, the authors develop four new families of such transfinite invariants for arbitrary closed orientable \(3\)-manifolds \(M\) and view these invariants to be a solution to Milnor's problem.\N\NOne such family \(\{{\bar {\mu}_\kappa}(M)\}\), indexed by ordinals, includes the classical Milnor invariants as a special case. The authors show the following: (i) the \({\bar {\mu}_\kappa}(M)\) inductively determine the isomorphism classes of the lower central series quotients (where induction extends to transfinite ordinals), (ii) the \({\bar {\mu}_\kappa}(M)\) are invariant under homology cobordism, (iii) the \({\bar {\mu}_\kappa}(M)\) with finite \(\kappa\) determine Milnor's link invariants (when \(M\) is \(0\)-surgery on a link in \(S^3\)), (iv) \({\bar {\mu}_\kappa}(M)\) is an obstruction to building gropes, (v) the \({\bar {\mu}_\kappa}(M)\) live in a certain explicitly characterized set \(S\) and every element of \(S\) is realized as \({\bar {\mu}_\kappa}(M)\) for some closed \(3\)-manifold \(M\).\N\NThe authors compute and analyze the invariants for certain torus bundles over \(S^1\). In particular, they show that there are infinitely many such \(3\)-manifolds \(M\) with vanishing \({\bar {\mu}_\kappa}(M)\) for all finite \(\kappa\), but having non-vanishing pairwise distinct \({\bar {\mu}_\omega}(M)\) for the first transfinite ordinal \(\omega\).