On Bousfield's problem for solvable groups of finite Prüfer rank

2017/04/07 by Ivanov, Sergei O.
#Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1704.02212

Abstract

For a group G and R=\mathbb Z,\mathbb Z/p,\mathbb Q we denote by G_R the R-completion of G. We study the map H_n(G,K)→ H_n( G_R,K), where (R,K)=(\mathbb Z,\mathbb Z/p),(\mathbb Z/p,\mathbb Z/p),(\mathbb Q,\mathbb Q). We prove that H₂(G,K)→ H₂( G_R,K) is an epimorphism for a finitely generated solvable group G of finite Prüfer rank. In particular, Bousfield's HK-localisation of such groups coincides with the K-completion for K=\mathbb Z/p,\mathbb Q. Moreover, we prove that H_n(G,K)→ H_n( G_R,K) is an epimorphism for any n if G is a finitely presented group of the form G=M\rtimes C, where C is the infinite cyclic group and M is a C-module.

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