Links, pictures and the homology of nilpotent groups

Authors
Citation
K. Igusa et Ke. Orr, Links, pictures and the homology of nilpotent groups, TOPOLOGY, 40(6), 2001, pp. 1125-1166
Citations number
44
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
TOPOLOGY
ISSN journal
0040-9383 → ACNP
Volume
40
Issue
6
Year of publication
2001
Pages
1125 - 1166
Database
ISI
SICI code
0040-9383(200111)40:6<1125:LPATHO>2.0.ZU;2-J
Abstract
We give a geometric slice-like characterization for the vanishing of Milnor 's link invariants by proving the k-slice conjecture. This conjecture state s that a link L has vanishing Milnor p-invariants of length less than or eq ual to 2k if and only if L bounds disjoint surfaces in a four disk in such a way that the fundamental group of the complement admits free nilpotent qu otients of class k. In the course of our proof, we compute the dimension le ss than or equal to 3 homology groups of finitely generated free nilpotent Lie rings and groups. We develop a new algorithm for constructing a weighte d chain resolution for a nilpotent group with torsion free lower central se ries quotients, and with the property that its associated graded complex is the Koszul complex of the associated graded Lie ring. This give a new deri vation of the May spectral sequence relating the group homology of the nilp otent group to the Lie ring homology of its associated graded Lie ring. Fin ally, we define Tt-invariants of "pictures" and use these to describe a gen erating set of cocycles in the cohomology of the free nilpotent groups. Som e sample computations follow. (C) 2001 Elsevier Science Ltd. All rights res erved.