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

Question

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!

Answer 1

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 $6g-7+2n+3b$.