If $M\_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1\_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:


$M^1\_{g} \rightarrow M\_{g}$


the fiber over a point $S \in M\_{g}$ is the sphere bundle associated to the tangent bundle of Riemann surface $S$?


I don't have appropriate coordinates that is probably why I can't show it.


I am asking this question because I want to show that if $M^{m,b}\_{g,n}$ is the moduli space of Riemann surfacw of genus $g$ and $b$ boundary components, and with $n$ interior marked points, $m=(m\_1,m\_2,\ldots,m\_{b})$ marked points on the boundary, $m\_i$ marked points on the $i$th boundary component, then the homology dimension


$H\_i(M^{m,b}\_{g,n},Q)=0,$ for $i\ge 6g-7+2n+3b+m$ possibly excludes some bad low dimensional cases.


I learned one way is tp assoicate it to mapping class group (especially we got a virtual cohomological dimension of mapping class group by Harer"on virtual cohomological dimension of mapping class group of oriented surface") and homology of mapping class group is the same as homology of the above moduli space, but I still don't understand how it works.
Thank you for help.



Question related to the moduli space of Riemann surfaces and a fibration
I'll give you references for the appropriate fact about the mapping class group. Let $Mod\_{g,b}^p$ be the mapping class group of a genus $g$ surface with $b$ boundary components and $p$ punctures $\Sigma\_{g,b}^p$. There are really two fact. The first is the Birman exact sequence (by the way, I believe this was Joan Birman's thesis!). It says that there is a short exact sequence
$$1 \longrightarrow \pi\_1(\Sigma\_{g,b}^{p-1}) \longrightarrow Mod\_{g,b}^p \longrightarrow Mod\_{g,b}^{p-1} \longrightarrow 1$$
except in the degenerate cases where either $g=0$ and $b+p$ is at most $3$, or $g=1$ and $b+p$ is at most $1$. 


This was originally proven in her paper


J. S. Birman,
Mapping class groups and their relationship to braid groups,
Comm. Pure Appl. Math. {\bf 22} (1969), 213--238.


Two other nice sources for it are Birman's book "Braids, Links, and Mapping Class Groups" and Farb-Margalit's book "A Primer on Mapping Class Groups", the latter of which is available [here](http://www.math.utah.edu/~margalit/primer/).


The second is a sort of variant on it, which says that (except in the degenerate cases above) we have a short exact sequence
$$1 \longrightarrow \pi\_1(U\Sigma\_{g,b-1}^p) \longrightarrow Mod\_{g,b}^p \longrightarrow Mod\_{g,b-1}^p \longrightarrow 1.$$
Here $U\Sigma\_{g,b-1}^p$ is the unit tangent bundle of $\Sigma\_{g,b-1}^p$.


It's a little harder to give a reference for this; the only place I know where it is proven is in section 3 of the following paper:


D. Johnson,
The structure of the Torelli group. I. A finite set of generators for ${\cal I}$,
Ann. of Math. (2) {\bf 118} (1983), no.~3, 423--442.


By the way, this also happens to be the earliest appearance of this in the literature that I know of, though I've seen it many times subsequently with no reference to Johnson's work and no proof.


The proof makes use of the Birman exact sequence above.


