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

Details Finite-type invariants of three-manifolds and the dimension subgroup problem Finite-type invariants of three-manifolds and the dimension subgroup problem

2006-05-18

arxiv.org/abs/math/0605497v3

J. London Math. Soc. 75:3 (2007) 791-811

Mathematics

Geometric Topology

20 pages, 7 figures. Minor changes in this final version

Scientific paper

For a certain class of compact oriented 3-manifolds, M. Goussarov and K. 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 analogue 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.

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