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A finite $\mathbb{Q}$–bad space

Sergei O Ivanov and Roman Mikhailov

Geometry & Topology 23 (2019) 1237–1249
Abstract

We prove that, for a free noncyclic group F, the second homology group H2(F̂, ) is an uncountable –vector space, where F̂ denotes the –completion of F. This solves a problem of A K Bousfield for the case of rational coefficients. As a direct consequence of this result, it follows that a wedge of two or more circles is –bad in the sense of Bousfield–Kan. The same methods as used in the proof of the above result serve to show that H2(F̂, ) is not a divisible group, where F̂ is the integral pronilpotent completion of F.

Keywords
homology, nilpotent completion, Bousfield–Kan completion, R–good space, R–bad space
Mathematical Subject Classification 2010
Primary: 14F35, 16W60, 55P60
References
Publication
Received: 27 July 2017
Revised: 29 July 2018
Accepted: 29 September 2018
Published: 28 May 2019
Proposed: Mark Behrens
Seconded: Haynes R Miller, Stefan Schwede
Authors
Sergei O Ivanov
Laboratory of Modern Algebra and Applications
St. Petersburg State University
Saint Petersburg
Russia
Roman Mikhailov
Laboratory of Modern Algebra and Applications
St. Petersburg State University
Saint Petersburg
Russia
St. Petersburg Department of Steklov Mathematical Institute
Saint Petersburg
Russia