TITLE: Cohomological dimension of a group
QUESTION [6 upvotes]: What is an intuitive, straightforward explanation of the cohomological dimension of a group ?
How does one compute the cohomological dimension of a group ?
Is there a good reference that explains this concept and provides examples ?

I am particularly interested in the cohomological dimension of the braid group $\mathcal{B}_n$ and some of its subgroups.

REPLY [3 votes]: Recall that group cohomology of $G$ are just cohomology of topological space $BG$ (in general, with coefficients in some local system). In particular, if there is a model of $BG$ that is $n$-dimensional CW-complex, then cohomological dimension of $G$ is $\le n$.
Now, there is a very nice model for $BB_n$: the space of $n$ (indistinguishable distinct) points of $\mathbb C$. Vieta map identify this space with $\mathbb C^n\setminus\Delta$ (where $\Delta$ is some kind of discriminant subset). In particular, cohomological dimension of $B_n$ is $\le 2n-1$ (oh... I think, the answer is, in fact, more like $n$). Canonical reference here is Arnold's “On some topological invariants of algebraic functions”, AFAIR.