We 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, that of spheres with four marked points, and that of tori with two marked points. We then prove the linear independence of the cycles in the rational homology groups in order to improve Wolpert's estimates for even degree Betti numbers of the moduli space of stable curves.
Kiyoshi OHBA. "Higher cycles on the moduli space of stable curves." J. Math. Soc. Japan 52 (2) 231 - 267, April, 2000. https://doi.org/10.2969/jmsj/05220231