Right exact group completion as a transfinite invariant of homology equivalence

Abstract

We consider a functor from the category of groups to itself, $G\mapsto {\mathbb{Z}}_{\infty }G$, that we call right exact $\mathbb{Z}$–completion of a group. It is connected with the pro-nilpotent completion $Ĝ$ by the short exact sequence $1\to {\underleftarrow{lim}}^1{M}_{n}G\to {\mathbb{Z}}_{\infty }G\to Ĝ\to 1$, where $M_{n}G$ is $n^{\text{ th}}$ Baer invariant of $G$. We prove that ${\mathbb{Z}}_{\infty }\left({\pi }_{1}X\right)$ is an invariant of homological equivalence of a space $X$. Moreover, we prove an analogue of Stallings’ theorem: if $G\to G^{\prime }$ is a $2$–connected group homomorphism, then ${\mathbb{Z}}_{\infty }G\cong {\mathbb{Z}}_{\infty }{G}^{\prime }$. We give examples of $3$–manifolds $X$ and $Y$ such that ${\widehat{\pi }}_{1}X\cong {\widehat{\pi }}_{1}Y$ but ${\mathbb{Z}}_{\infty }{\pi }_{1}X\ncong {\mathbb{Z}}_{\infty }{\pi }_{1}Y$. We prove that for a group $G$ with finitely generated $H_{1}G$ we have $\left({\mathbb{Z}}_{\infty }G\right)∕{\gamma }_{\omega }=Ĝ$. So the difference between $Ĝ$ and ${\mathbb{Z}}_{\infty }G$ lies in ${\gamma }_{\omega }$. This allows us to treat ${\mathbb{Z}}_{\infty }{\pi }_{1}X$ 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.