Higher cycles on the moduli space of stable curves (Q1574799)

Denote by \(\overline{M}_g\) the moduli space of stable curves of genus \(g\) \(\geq 3.\) It is known that \(\overline{M}_g\) is a compactification of the classical moduli space \(M_g\) of Riemann surfaces of genus \(g\) and that \(\overline{M}_g\) is a complex V-manifold of dimension \(3g-3,\) and that the compactification locus \(D=\overline{M}_g\backslash M_g,\) is the set of points in \(\overline{M}_g\) represented by stable curves with nodes. The purpose of this paper is to construct a number of analytic cycles on the moduli space of stable curves by using three moduli spaces: the moduli space of tori with one marked point, of spheres with four marked points, and of tori with two marked points, and to prove the linear independence of the cycles in the rational homology groups. Moreover, a better estimate for each Betti number of \(\overline{M}_g\) of even degree is given to improve Wolpert's estimate. Fix a Riemann surface \(S\) of genus \(g\) and a set of \(3g-3\) cutting curves on \(S\) for a certain pants decomposition of \(S.\) For \(k\leq 2g-2,\) the author defines a \(k\)-selection to be a selection \(\sigma \) of \(k\) from the \(3g-3\) cutting curves satisfying two conditions. Denote by \(\alpha _{g,k}\) the number of conjugacy classes of \(k\)-selections. Theorem A. When \(k\geq 2,\) \[ b_{2k}(\overline{M}_g)=b_{6g-6-2k}(\overline{M}_g)\geq \max (\alpha _{g,k},\alpha _{g,3g-3-k}). \] One of the additional results to supplement Theorem A is \[ \alpha _{g,k}>\frac 12 ( g-1 k) +\frac 12\sum_{k^{''},l}( k^{''}-1 l) ( k-2k^{''}+1 l+1) ( g-l-2 k-k^{''}). \]