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Let $\mathbf B_n$ be the braid group on $n$ strings.

What is known about the cohomology of $\mathbf B_n$ with coefficients in its integral group ring: $H^*(\mathbf B_n;\mathbb Z \mathbf B_n)$?

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The braid groups $B_n$ are Bieri-Eckmann duality groups of dimension $n-1$. It follows (either by definition or by a standard result depending on how you set things up) that $H^k(B_n;\mathbb{Z}[B_n])$ is $0$ for $k \neq n-1$ and is torsion-free for $k=n-1$.

Here is a brief description of how to see this. First, the group $B_n$ are of type F (i.e. they have compact Eilenberg-MacLane spaces). It follows that it is enough to find a finite-index subgroup of $B_n$ that is a duality group of dimension $n-1$. The pure braid group $PB_n$ do the job. To see that $PB_n$ is a duality group of dimension $n-1$, we use induction on $n$. The base case $n=1$ is trivial since $PB_1$ is the trivial group. For $n>1$, we will use the standard short exact sequence $$1 \longrightarrow F_{n-1} \longrightarrow PB_n \longrightarrow PB_{n-1} \longrightarrow 1,$$ where $F_{n-1}$ is the free group on $(n-1)$ generators and the map $PB_n \rightarrow PB_{n-1}$ comes from deleting the final strand. By induction $PB_{n-1}$ is a duality group of dimension $(n-2)$, and it is standard that $F_{n-1}$ is a duality group of dimension $1$. It follows that $PB_n$ is a duality group of dimension $(n-2)+1 = n-1$.

There is a more general theorem of Harer that says that all mapping class groups are virtual duality groups (and are actually duality groups if they are torsion-free), but the above argument is much easier than what Harer did.

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  • $\begingroup$ Hi Andy, thank you for a very nice answer. What is the situation for the commutator subgroup $[B_n,B_n]$? $\endgroup$ – Jarek Kędra Oct 15 '17 at 10:15
  • $\begingroup$ @JarekKędra: Sorry, I don't know much about the cohomology of $[B_n,B_n]$. Indeed, I don't even know what kind of finiteness properties it has, though I'm sure this must be in the literature somewhere. $\endgroup$ – Andy Putman Oct 15 '17 at 18:10
  • $\begingroup$ In algebraic geometry vanishing theorems (ie. theorems to show conditions for vanishing of sheaf cohomology groups) are useful to compute invariants, is there some particular application to showing conditions under which cohomology of the braid groups always vanishes for coefficients in a particular ring ie. what would the wider use of a 'vanishing theorem' in this case? $\endgroup$ – Hollis Williams May 4 at 17:10

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