Z-torsion homology for groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:15:22Z http://mathoverflow.net/feeds/question/77958 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77958/z-torsion-homology-for-groups Z-torsion homology for groups dpanagop 2011-10-12T21:30:06Z 2011-10-13T03:41:57Z <p>I would like to ask the following: if for a group $G$ the homology $H_n(G,\mathbb{Z})$ is $\mathbb{Z}$-torsion for every $n\geq n_0$, then what can be said concerning $\mathbb{Z}$-torsion for $H_k(G,M)$ where M is a $\mathbb{Z}G$-module? For example I know that if M is a trivial $G$-module, then $H_n(G,M)\simeq H_n(G,\mathbb{Z})\otimes_{\mathbb{Z}}M\oplus Tor_1^{\mathbb{Z}}(H_{n-1}(G,\mathbb{Z}),M)$ [Weibel, Th. 6.1.12] and hence $H_n(G,M)$ is $\mathbb{Z}$-torsion if $H_n(G,\mathbb{Z})$ is. What happens if M is not a trivial $G$-module?</p> <p>Weibel, "An introduction to homological Algebra"</p> http://mathoverflow.net/questions/77958/z-torsion-homology-for-groups/77960#77960 Answer by Chris Gerig for Z-torsion homology for groups Chris Gerig 2011-10-12T21:37:07Z 2011-10-12T21:37:07Z <p>This Kunneth formula still holds, I proved it here:</p> <p><a href="http://mathoverflow.net/questions/75472/kuenneth-formula-for-group-cohomology-with-nontrivial-action-on-the-coefficient/75485#75485" rel="nofollow">http://mathoverflow.net/questions/75472/kuenneth-formula-for-group-cohomology-with-nontrivial-action-on-the-coefficient/75485#75485</a></p> <p>which holds in our situation for all nontrivial $G$-modules.</p> http://mathoverflow.net/questions/77958/z-torsion-homology-for-groups/77991#77991 Answer by Ian Agol for Z-torsion homology for groups Ian Agol 2011-10-13T03:41:57Z 2011-10-13T03:41:57Z <p>There exists an acyclic group $G$ which has the property that there exists a finite-index normal subgroup $H\lhd G$ such that $\mathbb{Z} \leq H_1(H;\mathbb{Z})$. In particular, then $H_1(G;\mathbb{Z}[G/H]) \cong H_1(H;\mathbb{Z})$ is not torsion (by Shapiro's lemma). </p> <p>The example is the <a href="http://www.math.nus.edu.sg/~matberic/wild03923.pdf" rel="nofollow">fundamental group of the complement of a wild arc</a>. Examples in the linked paper, such as "Fox's stitch", are infinite cyclic covers of a hyperbolic 2-component link complement. There are such examples which are arithmetic, and therefore have a finite-index subgroup which is "RFRS", by a <a href="http://front.math.ucdavis.edu/0707.4522" rel="nofollow">theorem of mine</a>. Thus, for any element in the group, there is a finite-index subgroup for which it is homologically non-trivial. This property passes to subgroups, in which case the fundamental group of the complement of Fox's stitch has the property that there is a finite-index subgroup with infinite abelianization (in fact, a non-trivial homomorphism to $\mathbb{Z}$). </p>