Pontrjagin-Thom maps and the homology of the moduli stack of stable curves
arXiv:0712.0702
Abstract
We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack.
30 pages, 3 figures - v2: expanded material on homotopy types of stacks, extended Pontrjagin-Thom construction to all local quotient stacks, added references