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Right exact group completion as a transfinite invariant of homology equivalence

Sergei O Ivanov and Roman Mikhailov

Algebraic & Geometric Topology 21 (2021) 447–468

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 1 lim1MnG G Ĝ 1, 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 G G is a 2–connected group homomorphism, then GG. We give examples of 3–manifolds X and Y such that π̂1Xπ̂1Y but π1X π1Y . 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.

links, $3$–manifolds, concordance, homology cobordism, pro-nilpotent completion, localization, $\mu$–invariants
Mathematical Subject Classification 2010
Primary: 55P60
Received: 26 September 2019
Revised: 28 February 2020
Accepted: 11 May 2020
Published: 25 February 2021
Sergei O Ivanov
Laboratory of Modern Algebra and Applications
St Petersburg State University
Saint Petersburg
Roman Mikhailov
St Petersburg Department of Steklov Mathematical Institute
Laboratory of Modern Algebra and Applications
St Petersburg State University
Saint Petersburg