Cohomological dimension of a group acting on a cellular complex - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:14:41Z http://mathoverflow.net/feeds/question/38844 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38844/cohomological-dimension-of-a-group-acting-on-a-cellular-complex Cohomological dimension of a group acting on a cellular complex Romeo 2010-09-15T16:33:04Z 2010-09-15T21:11:30Z <p>Let $G$ be a group acting on $X$, $X$ a cellular complex and $cd(G)$ the cohomological dimension of $G$.</p> <p>2 things:</p> <p>(1) I'm looking for a reference (or proof!) of this:</p> <p>Suppose $X$ is acyclic. Then $cd(G) \leq max_{\sigma} \space cd(Stab(\sigma) + dim \space \sigma$, where $\sigma$ runs over the cells of $X$.</p> <p>(2) If $X$ isn't acyclic, can anything be said about $cd(G)$?</p> http://mathoverflow.net/questions/38844/cohomological-dimension-of-a-group-acting-on-a-cellular-complex/38849#38849 Answer by Bruce Ikenaga for Cohomological dimension of a group acting on a cellular complex Bruce Ikenaga 2010-09-15T17:33:09Z 2010-09-15T17:33:09Z <p>Reference: Serre, J-P.(1971) "Cohomologie des groupes discrets," Ann. Math. Studies 70, 77-169 (Proposition 11, page 93). Serre credits this to Quillen, and I've never succeeded in locating this in any of Quillen's papers. Does anyone know where it is?</p> <p>Proof: Use the equivariant cohomology spectral sequence.</p> http://mathoverflow.net/questions/38844/cohomological-dimension-of-a-group-acting-on-a-cellular-complex/38854#38854 Answer by Nikita for Cohomological dimension of a group acting on a cellular complex Nikita 2010-09-15T18:22:40Z 2010-09-15T18:22:40Z <p>If you don't require $X$ to be acyclic, then you can't say anything. Indeed, every group acts freely on its <a href="http://en.wikipedia.org/wiki/Cayley_graph" rel="nofollow">Cayley graph</a>, which is a 1-dimensional cell complex. </p> <p>You're not even saved by assuming the $X$ is highly connected. By attaching cells to the Cayley graph in an equivariant manner, you can obtain a $k$-connected $(k+1)$-dimensional complex on which the group acts freely.</p> http://mathoverflow.net/questions/38844/cohomological-dimension-of-a-group-acting-on-a-cellular-complex/38859#38859 Answer by Mark Grant for Cohomological dimension of a group acting on a cellular complex Mark Grant 2010-09-15T18:57:06Z 2010-09-15T21:11:30Z <p>If $X$ is a free contractible $G$-complex, this is the <a href="http://www.ams.org/mathscinet-getitem?mr=0085510" rel="nofollow">Eilenberg-Ganea Theorem</a></p>