#### Volume 21, issue 1 (2021)

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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
##### Abstract

We consider a functor from the category of groups to itself, $G↦{ℤ}_{\infty }G\phantom{\rule{-0.17em}{0ex}}$, that we call right exact $ℤ$–completion of a group. It is connected with the pro-nilpotent completion $Ĝ$ by the short exact sequence $1\to {\underset{←}{lim}}^{1}{M}_{n}G\to {ℤ}_{\infty }G\to Ĝ\to 1$, where ${M}_{n}G$ is Baer invariant of $G\phantom{\rule{-0.17em}{0ex}}$. We prove that ${ℤ}_{\infty }\left({\pi }_{1}X\right)$ is an invariant of homological equivalence of a space $X\phantom{\rule{-0.17em}{0ex}}$. Moreover, we prove an analogue of Stallings’ theorem: if $G\to {G}^{\prime }$ is a $2$–connected group homomorphism, then ${ℤ}_{\infty }G\cong {ℤ}_{\infty }{G}^{\prime }\phantom{\rule{-0.17em}{0ex}}$. We give examples of $3$–manifolds $X$ and $Y$ such that ${\stackrel{̂}{\pi }}_{1}X\cong {\stackrel{̂}{\pi }}_{1}Y$ but ${ℤ}_{\infty }{\pi }_{1}X\ncong {ℤ}_{\infty }{\pi }_{1}Y\phantom{\rule{-0.17em}{0ex}}$. We prove that for a group $G$ with finitely generated ${H}_{1}G$ we have $\left({ℤ}_{\infty }G\right)∕{\gamma }_{\omega }=Ĝ\phantom{\rule{-0.17em}{0ex}}$. So the difference between $Ĝ$ and ${ℤ}_{\infty }G$ lies in ${\gamma }_{\omega }$. This allows us to treat ${ℤ}_{\infty }{\pi }_{1}X$ as a transfinite invariant of $X\phantom{\rule{-0.17em}{0ex}}$. The advantage of our approach is that it can be used not only for $3$–manifolds but for arbitrary spaces.

##### Keywords
links, $3$–manifolds, concordance, homology cobordism, pro-nilpotent completion, localization, $\mu$–invariants
Primary: 55P60