Cohomological dimension of a group acting on a cellular complex - MathOverflow

Cohomological dimension of a group acting on a cellular complex

2010-09-15T16:33:04Z

Let $G$ be a group acting on $X$, $X$ a cellular complex and $cd(G)$ the cohomological dimension of $G$.

2 things:

(1) I'm looking for a reference (or proof!) of this:

Suppose $X$ is acyclic. Then $cd(G) \leq max_{\sigma} \space cd(Stab(\sigma) + dim \space \sigma$, where $\sigma$ runs over the cells of $X$.

(2) If $X$ isn't acyclic, can anything be said about $cd(G)$?

Answer by Bruce Ikenaga for Cohomological dimension of a group acting on a cellular complex

2010-09-15T17:33:09Z

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?

Proof: Use the equivariant cohomology spectral sequence.

Answer by Nikita for Cohomological dimension of a group acting on a cellular complex

2010-09-15T18:22:40Z

If you don't require $X$ to be acyclic, then you can't say anything. Indeed, every group acts freely on its Cayley graph, which is a 1-dimensional cell complex.

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.

Answer by Mark Grant for Cohomological dimension of a group acting on a cellular complex

2010-09-15T18:57:06Z

If $X$ is a free contractible $G$-complex, this is the Eilenberg-Ganea Theorem