Narrow Results

For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y₂-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group.

In this paper, when Σ has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.

Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey, and Friedl are generally factorized into the part of the Magnus matrix and that of a certain Reidemeister torsion, both of which are known as invariants of homology cylinders over a surface. In this paper, we study more details of the invariants and give their concrete calculations after restricting to the case of the invariants associated with metabelian quotients of knot groups. We provide explicit computational results of the invariants for all the 12-crossings non-fibered homologically fibered knots.

For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does not normally generate $F$. The question comes from a study of Hirzebruch-type invariants from iterated $p$-covers for three-dimensional homology cylinders.