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Right exact group completion as a transfinite invariant of the homology equivalence
- Source :
- Algebr. Geom. Topol. 21 (2021) 447-468
- Publication Year :
- 2019
-
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.
- Subjects :
- Mathematics - Algebraic Topology
Subjects
Details
- Database :
- arXiv
- Journal :
- Algebr. Geom. Topol. 21 (2021) 447-468
- Publication Type :
- Report
- Accession number :
- edsarx.1909.10181
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.2140/agt.2021.21.447