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2021 Right exact group completion as a transfinite invariant of homology equivalence
Sergei Ivanov, Roman Mikhailov
Algebr. Geom. Topol. 21(1): 447-468 (2021). DOI: 10.2140/agt.2021.21.447

Abstract

We consider a functor from the category of groups to itself, GG, that we call right exact –completion of a group. It is connected with the pro-nilpotent completion Ĝ by the short exact sequence 1lim1MnGGĜ1, where MnG is n th Baer invariant of G. We prove that (π1X) is an invariant of homological equivalence of a space X. Moreover, we prove an analogue of Stallings’ theorem: if GG is a 2–connected group homomorphism, then GG. We give examples of 3–manifolds X and Y such that π̂1Xπ̂1Y but π1Xπ1Y. We prove that for a group G with finitely generated H1G we have (G)γω=Ĝ. So the difference between Ĝ and G lies in γω. This allows us to treat π1X as a transfinite invariant of X. The advantage of our approach is that it can be used not only for 3–manifolds but for arbitrary spaces.

Citation

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Sergei Ivanov. Roman Mikhailov. "Right exact group completion as a transfinite invariant of homology equivalence." Algebr. Geom. Topol. 21 (1) 447 - 468, 2021. https://doi.org/10.2140/agt.2021.21.447

Information

Received: 26 September 2019; Revised: 28 February 2020; Accepted: 11 May 2020; Published: 2021
First available in Project Euclid: 16 March 2021

Digital Object Identifier: 10.2140/agt.2021.21.447

Subjects:
Primary: 55P60

Keywords: $\mu$–invariants , $3$–manifolds , concordance , Homology cobordism , links , Localization , pro-nilpotent completion

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.21 • No. 1 • 2021
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