On the rank, the deficiency and the homological dimension of groups: The computation of a lower bound via fox ideals

Abstract

For any group G we introduce natural analogues Im(G)<ℤ G of the Fox ideal I1(G) for all dimensions m>1. These are shown to have the following applications:§1: Non-trivial representations of ℤ G/Im(G) as matrices over a commutative ring yield lower bounds for (a) the rank of G (for m=1), (b) the deficiency of G (for m=2), and (c) the homological dimension of G (for m<hdim(G)).§2: A "Whitehead group" ℝm(G) is defined over the quotient ring ℤ G/Im(G) which measures whether maps between finite CW-complexes are simple-homotopy equivalences. A formula is presented which describes a decomposition of ℝm(G) in the case where G is a free product.