An extension of Milnorʹs \̄gm-invariants

We study embeddings in a certain fixed, nontrivial homotopy class of one copy of the circle S1 in any closed, aspherical, orientable, irreducible 3-dimensional Seifert fibered 3-manifold, M, and extract a collection of numerical concordance invariants from certain quotients of the fundamental group of the complement of the knot. (The property of the fundamental group of the target 3-manifold that is needed to produce these invariants is precisely the one that guarantees that the manifold is a Seifert fibered space, by the recently proved Seifert fibered space theorem.) These extensions of Milnorʹs \̄gm-invariants detect “self-linking” phenomena that are nonsimply connected analogues to the “higher order” linking phenomena detected by the classical \̄gm-invariants. In particular, they are obstructions to an embedding being concordant to a characteristic embedding. Examples of knots that realize some of these invariants are constructed for M = T3, and a realization theorem is proved for any M, using a generic, ordinary Seifert fiber as the model of a “trivial” embedding.