Abstract
We study the singular homology (with field coefficients) of the moduli stack \({\overline{\mathfrak{M}}_{g, n}}\) of stable n-pointed complex curves of genus g. Each irreducible boundary component of \({\overline{\mathfrak{M}}_{g, n}}\) determines via the Pontrjagin–Thom construction a map from \({\overline{\mathfrak{M}}_{g, n}}\) 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 detects many new torsion classes in the homology of \({\overline{\mathfrak{M}}_{g, n}}\).
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Ebert, J., Giansiracusa, J. Pontrjagin–Thom maps and the homology of the moduli stack of stable curves. Math. Ann. 349, 543–575 (2011). https://doi.org/10.1007/s00208-010-0518-2
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DOI: https://doi.org/10.1007/s00208-010-0518-2