## Homology dimension of the mapping class group of a surface with boundary

There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is $H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except $(g,n)=(0,3),(0,2),(0,1),(0,0),(1,1)$. This result (see Costello: Gromov-Witten potential associated to a TCFT) can be derived from the virtual cohomology dimension of the mapping class group (see J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math). I am wondering if there is such a theorem for surfaces with boundary. For example, is there any similar result for mapping class groups of orientable surfaces with boundary(and marked points if necessary)? Then can we get a result similar to the above dimension bound for moduli spaces of Riemann surfaces with boundary and marked points? I just want to know if such a result already exists.

Can we use reduced homology so that we don 't need exclude those low dimensional cases? I mean if it is natural?Thanks!

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There is a fibration sequence $$\mathbb{S}(\Sigma_g) \to \mathcal{M}_{g}^1 \to \mathcal{M}_g$$ where $\mathcal{M}_g$ is the moduli space of Riemann surfaces, $\mathcal{M}_{g}^1$ is the moduli space of Riemann surfaces with a single boundary, and $\mathbb{S}(\Sigma_g)$ is the sphere bundle associated to the tangent bundle of a surface of genus $g$.
As the homology of $\mathcal{M}_g$ vanishes in degrees at least $6g-7$, and $\mathbb{S}(\Sigma_g)$ is a 3-manifold, the Serre spectral sequence implies that the homology of $\mathcal{M}_{g}^1$ vanishes in degrees at least $6g-4$.
Iterating, the homology of $\mathcal{M}_{g, n}^b$ vanishes in degrees at least $6g-7+2n+3b$.
 Hi,Oscar, according to your argument, it seems that I can show the homology of $M^{b,m}_{g,n}$ (here, there could be marked points on the boundary,for which $m=(m_1,m_2,\ldots,m_b), m_i$ is the number of marked points on the $i$th boundary component) vanishes in degrees at least $6g-7+2n+3b+m$. However, when $6g-7+2n+3b+m=0$,$H_0$ will not be 0. Is this result true for other cases except the cases for $6g-7+2n+3b+m=0$? In fact, for the homology dimension for closed surface we need to exclude the case $(g,n)=(0,3)$ because $H_0$ is also not 0. How to deal with such things? Thanks! – HYYY Jul 30 2010 at 13:10 If you take reduced homology this problem does not appear. – Oscar Randal-Williams Jul 30 2010 at 15:09 Hi,Oscar,thank you for your help. but could you show me why that is a sphere bundle assoicated to the tangent bundle? and is that fibration just a fiber bundle?Thanks! – HYYY Aug 7 2010 at 10:50