First some definitions: $\bar{M_{g,n}}$ is Deligne-Mumford space, i.e., the moduli space of stable nodal complex projective curves of genus $g$ with $n$ marked points. It is a complex orbifold, $\partial \bar{M_{g,n}}$ is the locus in $\bar{M_{g,n}}$ corresponding to nodal curves (with singularity). Do we have any relationship between the homology groups $H_{*}(M_{g,n},Q)$ and the homology groups of $\bar{M_{g,n}}$, $\partial \bar{M_{g,n}}$, the pair $(\bar{M_{g,n}},\partial \bar{M_{g,n}})$ (relative homology); here $M_{g,n}$ is $\bar{M_{g,n}}\setminus \partial \bar{M_{g,n}}$ the locus of smooth curves?

The point is for any pair of compact oriented manifolds $(X, Y), Y\subset X$, can we calculate the homology groups of $X\setminus Y$ from those of $X, Y$ and the relative homology groups $(X,Y)$ (it is not an excision case)?

This is a problem I find on page 23 of the paper: Costello, "Gromov-Witten potential associated to a TCFT", (although there it is $\bar{M_{g,n}}/S_n$, modulo the action of permutation of marked points, but it is not a big deal).

One more question is: is $\bar{M_{g,n}}/S_n$ orbifold?

  • $\begingroup$ I have taken the liberty of changing "/" to "\" (setminus) above, to avoid confusion with the quotient space. $\endgroup$ – Tom Church Jul 18 '10 at 19:11
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    $\begingroup$ There is the Poincar\'e-Lefschetz isomorphism $H^{\ast}(M_{g,n},\mathbf{Q})=H_{2d-\ast}(\bar M_{g,n},\partial \bar M_{g,n},\mathbf{Q})$ where $d$ is the complex dimension of the moduli space $M_{g,n}$. We can apply the Poincar\'e duality since the dualizing sheaf over the rationals is constant. Over the integers the situation is trickier since the stack cohomology is no longer isomorphic to the cohomology of the coarse moduli space and the dualizing sheaf of the latter is no longer constant. $\endgroup$ – algori Jul 18 '10 at 19:19
  • $\begingroup$ Hi, algori, could you explain "We can apply the Poincar\'e duality since the dualizing sheaf over the rationals is constant"a little more?Or is that isomorphism only for this special case (because Lefschetz duality normally require the left side to be$\bar{M_{g,n}}$. Is there any reference about the above isomorphism? $\endgroup$ – HYYY Jul 18 '10 at 19:32
  • $\begingroup$ Jeffrey -- if $U$ is an orientable manifold of real dimension $d$ and $X\supset$ is an arbitrary compactification, then $H^{\ast}(U)=H_{d-\ast}^{BM}(X)=H_{d-\ast}(X,\partial X)$ (arbitrary coefficients). Here $H^{BM}$ is the Borel-Moore homology. The first one of these isomorphisms is the Poinca\'e-Lefschetz duality; the second one is general. All this holds for orientable orbifolds provided one is working over the rationals. If $U$ is non-orientable, obvious modifications apply. $\endgroup$ – algori Jul 18 '10 at 23:48
  • $\begingroup$ that is, $X\supset U$ is an arbitrary compactification. It's so annoying that one can't edit comments. $\endgroup$ – algori Jul 18 '10 at 23:50

There is some general machinery that is perfectly suited to this question.

Consider the following setup: let $U$ be the complement of a normal crossing divisor $D$ in a compact complex manifold (or orbifold) $X$. (In the special case at hand, $U = \mathcal{M}_{g,n}$, and $X$ is the Deligne-Mumford compactification.) With a bit of work one can see that the Leray spectral sequence for the inclusion $U\hookrightarrow X$ has $E_2$ page given by \[ E^{p,q}_2 = \oplus_S H^{q-2p}(D_S;\mathbb{C}) \] where the sum runs over the closed boundary strata of codimension $p$ and the differential on this page is given by $da = \Sigma_T \pm (i_{S,T})_! a$ where $a\in H^{q-2p}(D_S;\mathbb{C})$ and $T$ runs over codimension $p-1$ boundary strata that contain $D_S$, and $(i_{S,T})_!$ is the pushforward along the inclusion $D_S \hookrightarrow D_T$. One can also get this spectral sequence from the weight filtration on the complex of forms with logarithmic poles along $D$.

Deligne proved that this spectral sequence degenerates at the $E_2$ page. So the cohomology of this page is the associated graded for $H^*(U;\mathbb{C})$.

In the case of the moduli space of curves and its Deligne-Mumford-Knudsen compactification, the $E_2$ page is described in terms of the cohomology of the various strata of the boundary, which are isomorphic to smaller compactified moduli spaces. There have been quite a few papers that used this spectral sequence to prove interesting things. For instance, there is an old paper of Voronov (alg-geom/9708019) (from just a couple of years before the proof of the Madsen-Weiss theorem) in which he analyzes this SS to show that the rational homotopy type of $\mathcal{M}_{g,n}$ is stable in the Harer-Ivanov stable range and moreover, it is formal in this range.

As an aside, very shortly after Voronov's paper, there was Tillmann's paper in which she showed that $\mathcal{M}_{g,n}$ has the homology of an infinite loop space in the stable range, which also implies rational homotopy formality in the stable range.


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