Right exact group completion as a transfinite invariant of homology equivalence

Ivanov, Sergei; Mikhailov, Roman

doi:10.2140/agt.2021.21.447

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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

DOI: 10.2140/agt.2021.21.447

Abstract

We consider a functor from the category of groups to itself, $G \mapsto ℤ_{\infty} G$ , that we call right exact $ℤ$ –completion of a group. It is connected with the pro-nilpotent completion $Ĝ$ by the short exact sequence $1 \to {lim_{\leftarrow}}^{1} M_{n} G \to ℤ_{\infty} G \to Ĝ \to 1$ , where $M_{n} G$ is $n^{th}$ Baer invariant of $G$ . We prove that $ℤ_{\infty} (π_{1} X)$ is an invariant of homological equivalence of a space $X$ . Moreover, we prove an analogue of Stallings’ theorem: if $G \to G^{'}$ is a $2$ –connected group homomorphism, then $ℤ_{\infty} G ≅ ℤ_{\infty} G^{'}$ . We give examples of $3$ –manifolds $X$ and $Y$ such that ${\hat{π}}_{1} X ≅ {\hat{π}}_{1} Y$ but $ℤ_{\infty} π_{1} X ≇ ℤ_{\infty} π_{1} Y$ . We prove that for a group $G$ with finitely generated $H_{1} G$ we have $(ℤ_{\infty} G) ∕ γ_{ω} = Ĝ$ . So the difference between $Ĝ$ and $ℤ_{\infty} G$ lies in $γ_{ω}$ . This allows us to treat $ℤ_{\infty} π_{1} X$ 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.

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Keywords

links, $3$–manifolds, concordance, homology cobordism, pro-nilpotent completion, localization, $\mu$–invariants

Mathematical Subject Classification 2010

Primary: 55P60

References

Publication

Received: 26 September 2019

Revised: 28 February 2020

Accepted: 11 May 2020

Published: 25 February 2021

Authors

Sergei O Ivanov
Laboratory of Modern Algebra and Applications St Petersburg State University Saint Petersburg Russia

Roman Mikhailov
St Petersburg Department of Steklov Mathematical Institute Laboratory of Modern Algebra and Applications St Petersburg State University Saint Petersburg Russia

Volume 21, issue 1 (2021)