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Homology and derived p ‐series of groups
Author(s) -
Cochran Tim,
Harvey Shelly
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn046
Subject(s) - modulo , mathematics , homomorphism , quotient , abelian group , central series , series (stratigraphy) , homology (biology) , pure mathematics , combinatorics , isomorphism (crystallography) , fundamental group , free group , group (periodic table) , discrete mathematics , crystallography , nilpotent , paleontology , biochemistry , chemistry , crystal structure , gene , nilpotent group , biology , organic chemistry
We give homological conditions that ensure that a group homomorphism induces an isomorphism modulo any term of the derived p − series , in analogy to Stallings's 1963 result for the p ‐lower central series. In fact, we prove a stronger theorem that is analogous to Dwyer's extensions of Stallings’ results. It follows that spaces that are ℤ p ‐homology equivalent have isomorphic fundamental groups modulo any term of their p ‐derived series. Various authors have related the ranks of the successive quotients of the p ‐lower central series and of the derived p ‐series of the fundamental group of a 3‐manifold M to the volume of M , to whether certain subgroups of π 1 ( M ) are free, to whether finite index subgroups of π 1 ( M ) map onto non‐abelian free groups, and to whether finite covers of M are ‘large’ in various other senses.