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edited Jul 5, 2011 at 21:28

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The other answers are correct, but I wanted to point out a quick way to see that $B_n$ has cohomological dimension $n-1$.

One obtains a lower bound of $n-1$ since $\mathbb{Z}^{n-1}$ is a subgroup of $B_n$. Take $n-1$ disjoint non-isotopic loops forming a pants decomposition of the $n$ punctured plane, then Dehn twists about these give a subgroup isomorphic to $\mathbb{Z}^{n-1}$.

For an upper bound, one may use the fact that the moduli space of $n$ points in $\mathbb{C}$ (normalized to have sum $=0$) is a Stein manifold of complex dimension $n-1$, and therefore has a spine of dimension $n-1$. The fundamental group of this space is $B_n$. This space is equivalent to the space of monic polynomials of degree $n$ with zero trace (coefficient of degree $n-1=0$) and non-zero discriminant, which is how one may see that it is Stein (actually, an affine variety). The fact that the moduli space is a $K(B_n,1)$ follows from Teichmuller theory, or one may pass to the finite-sheeted cover of $n$ marked points, and see that this is an iterated surface bundle, and therefore its universal cover is contractible.

The other answers are correct, but I wanted to point out a quick way to see that $B_n$ has cohomological dimension $n-1$.

One obtains a lower bound of $n-1$ since $\mathbb{Z}^{n-1}$ is a subgroup of $B_n$. Take $n-1$ disjoint non-isotopic loops forming a pants decomposition of the $n$ punctured plane, then Dehn twists about these give a subgroup isomorphic to $\mathbb{Z}^{n-1}$.

For an upper bound, one may use the fact that the moduli space of $n$ points in $\mathbb{C}$ (normalized to have sum $=0$) is a Stein manifold of complex dimension $n-1$, and therefore has a spine of dimension $n-1$. The fundamental group of this space is $B_n$. This space is equivalent to the space of monic polynomials of degree $n$ with zero trace (coefficient of degree $n-1=0$) and non-zero discriminant, which is how one may see that it is Stein (actually, an affine variety). The fact that the moduli space is a $K(B_n,1)$ follows from Teichmuller theory.

The other answers are correct, but I wanted to point out a quick way to see that $B_n$ has cohomological dimension $n-1$.

One obtains a lower bound of $n-1$ since $\mathbb{Z}^{n-1}$ is a subgroup of $B_n$. Take $n-1$ disjoint non-isotopic loops forming a pants decomposition of the $n$ punctured plane, then Dehn twists about these give a subgroup isomorphic to $\mathbb{Z}^{n-1}$.

For an upper bound, one may use the fact that the moduli space of $n$ points in $\mathbb{C}$ (normalized to have sum $=0$) is a Stein manifold of complex dimension $n-1$, and therefore has a spine of dimension $n-1$. The fundamental group of this space is $B_n$. This space is equivalent to the space of monic polynomials of degree $n$ with zero trace (coefficient of degree $n-1=0$) and non-zero discriminant, which is how one may see that it is Stein (actually, an affine variety). The fact that the moduli space is a $K(B_n,1)$ follows from Teichmuller theory, or one may pass to the finite-sheeted cover of $n$ marked points, and see that this is an iterated surface bundle, and therefore its universal cover is contractible.

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created Jul 5, 2011 at 21:18

68.8k
3
194
358

The other answers are correct, but I wanted to point out a quick way to see that $B_n$ has cohomological dimension $n-1$.

One obtains a lower bound of $n-1$ since $\mathbb{Z}^{n-1}$ is a subgroup of $B_n$. Take $n-1$ disjoint non-isotopic loops forming a pants decomposition of the $n$ punctured plane, then Dehn twists about these give a subgroup isomorphic to $\mathbb{Z}^{n-1}$.

For an upper bound, one may use the fact that the moduli space of $n$ points in $\mathbb{C}$ (normalized to have sum $=0$) is a Stein manifold of complex dimension $n-1$, and therefore has a spine of dimension $n-1$. The fundamental group of this space is $B_n$. This space is equivalent to the space of monic polynomials of degree $n$ with zero trace (coefficient of degree $n-1=0$) and non-zero discriminant, which is how one may see that it is Stein (actually, an affine variety). The fact that the moduli space is a $K(B_n,1)$ follows from Teichmuller theory.