Finite-type invariants of 3-manifolds and the dimension subgroup problem

Abstract

For a certain class of compact oriented 3-manifolds, Goussarov and Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of ‘cut-and-paste’ operations defined by the lower central series of the Torelli group of a surface. In this paper, we observe that this is a variation of a classical problem in group theory, namely the ‘dimension subgroup problem’. This viewpoint allows us to prove, by purely algebraic methods, an analog of the Goussarov–Habiro conjecture for finite-type invariants with values in a fixed field. We deduce that their original conjecture is true at least in a weaker form.