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.
partially supported by a 4 month visiting membership from M.S.R.I., Berkeley, and by a Grant from the German-Israeli Foundation for Scientific Research and Development (G.I.F.)
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© 1990 Springer-Verlag
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Lustig, M. (1990). On the rank, the deficiency and the homological dimension of groups: The computation of a lower bound via fox ideals. In: Latiolais, P. (eds) Topology and Combinatorial Group Theory. Lecture Notes in Mathematics, vol 1440. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084460
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DOI: https://doi.org/10.1007/BFb0084460
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