finite complex with non-finitely generated homology with local coefficients - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:07:45Z http://mathoverflow.net/feeds/question/101830 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101830/finite-complex-with-non-finitely-generated-homology-with-local-coefficients finite complex with non-finitely generated homology with local coefficients Ricardo Andrade 2012-07-10T07:33:13Z 2012-07-11T09:41:49Z <p>I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex $X$ such that some homology group with local coefficients $H_n(X,{\mathbb Z}[\pi_1 X])$ is not a finitely generated ${\mathbb Z}[\pi_1 X]$-module.</p> <p>Such an example would in particular give a finitely presented group $\pi$, and a chain complex of finitely generated free ${\mathbb Z}[\pi]$-modules whose homology groups are not all finitely generated over ${\mathbb Z}[\pi]$. Suggestions on finding an explicit example of such a chain complex (of length 3, without loss of generality) are also welcome.</p> http://mathoverflow.net/questions/101830/finite-complex-with-non-finitely-generated-homology-with-local-coefficients/101838#101838 Answer by HW for finite complex with non-finitely generated homology with local coefficients HW 2012-07-10T10:28:34Z 2012-07-11T09:41:49Z <p>As Ricardo points out in the comments, there's an error in my sketched calculation below. I also didn't notice the requirement that $X$ should be finite, so the natural $BK$ fails on two counts! However, it seems possible that a presentation complex for $K$ would do the job. Stallings shows that $\pi_2$ of any complex with $\pi_1=K$ is infinitely generated as a $K$-module.</p> <p>==========</p> <p>I think you want to start with a famous example of Stallings, from the paper '<a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=AUCN&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=stallings%252C%2520j%2a&amp;s5=&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=40&amp;mx-pid=158917" rel="nofollow">A finitely presented group whose 3-dimensional integral homology is not finitely generated</a>'. Stallings constructs a finitely presented group $K$ with the property that there is no projective resolution of $\mathbf{Z}$ over $\mathbf{Z}[K]$ which is finitely generated in dimension 3' (Corollary 1).</p> <p>In fact, as observed by Bieri, $K$ can be realizes as an explicit subgroup of the direct product of three free groups, $G=F_2\times F_2\times F_2$: $K$ is the kernel of a map $G\to\mathbf{Z}$ that sends every generator to $1$. So $K$ defines an explicit' 3-complex, namely a covering space of the natural $K(G,1)$. As $K$ is 3-dimensional and finitely presented, it follows from Corollary 1 that $H_3(K,\mathbf{Z}[K])$ is infinitely generated.</p> <p>Stallings's paper was the starting point for many beautiful constructions. Highlights include the word of <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&amp;pg1=CNO&amp;s1=390078&amp;loc=fromrevtext" rel="nofollow">Bieri</a> and <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=AUCN&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;r=1&amp;review_format=html&amp;s4=bestvina&amp;s5=brady&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq" rel="nofollow">Bestvina--Brady</a> .</p>