Right exact group completion as a transfinite invariant of the 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 pronilpotent completion $\hat{G}$ by the short exact sequence $1\to {\underleftarrow{\lim }}^1{M}_{n}G\to {\mathbb{Z}}_{\infty }G\to \hat{G}\to 1,$ 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 $G\to G^{\prime }$ is a 2-connected group homomorphism, then ${\mathbb{Z}}_{\infty }G\cong {\mathbb{Z}}_{\infty }{G}^{\prime }.$ We give examples of $3$-manifolds $X,Y$ 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 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.

References in corpus (1)

• Hidden torsion, 3-manifolds, and homology cobordism