Right exact group completion as a transfinite invariant of the homology equivalence

TL;DR

The paper introduces a right exact $bZ$-completion functor for groups, showing it as a transfinite invariant of spaces that refines the pronilpotent completion and is preserved under homological equivalences.

Contribution

It defines and studies the right exact $bZ$-completion of groups, establishing its invariance under homological equivalence and its relation to pronilpotent completion, extending invariants to arbitrary spaces.

Findings

$bZ_ty G$ is an invariant of homological equivalence.

$bZ_ty G$ is preserved under 2-connected homomorphisms.

Examples show $bZ_ty G$ distinguishes spaces with same pronilpotent completion.

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 pronilpotent completion $\hat G$ by the short exact sequence $1\to {\varprojlim}^1\: M_n G \to \mathbb Z_\infty G \to \hat G \to 1,$ where $M_n G$ is $n$-th Baer invariant of $G.$ We prove that $\mathbb Z_\infty \pi_1(X)$ 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 $\mathbb Z_\infty G\cong \mathbb Z_\infty G'.$ We give examples of $3$-manifolds $X,Y$ such that $ \hat{\pi_1(X)}\cong \hat{\pi_1( Y)}$ but $\mathbb Z_\infty \pi_1(X)\not \cong \mathbb Z_\infty \pi_1(Y).$ We prove that for a finitely generated group $G$ we have $(\mathbb Z_\infty G)/ \gamma_\omega= \hat G.$ So the difference between $\hat G$ 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.