MILNOR’S INVARIANTS FOR KNOTS AND LINKS IN CLOSED ORIENTABLE 3-MANIFOLDS

Abstract

In his 1957 paper, John Milnor introduced a collection of invariants for links in S3 detecting higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants, now known as Milnor’s µ-invariants, were later shown to be topological link concordance invariants and have since inspired decades of consequential research. Milnor’s invariants have many interpretations, and there have been numerous attempts to extend them to other settings. In this thesis, we extend Milnor’s invariants to topological concordance invariants of knots and links in general closed orientable 3-manifolds. These invariants unify and generalize all previous versions of Milnor’s invariants, including Milnor’s original invariants for links in S3.