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Pontrjagin-Thom maps and the homology of the moduli stack of stable curves. (English) Zbl 1257.32011
This paper is concerned with the study of the singular homology with field coefficients of the moduli stack \(\overline{\mathcal M}_{g,n}\) of Deligne-Mumford-Kudsen stable curves of genus \(g\) with \(n\) labelled marked points. Particular emphasis is given to the study of the mod \(p\) homology rather then the rational homology, as the first has received little attention when compared with the second. As the mod \(p\) homology of \(\overline{\mathcal M}_{g,n}\) and of its coarse moduli space are not necessarily isomorphic (as it is the case for the rational homology), the authors concentrate on the study of the homology of the stack rather then the space. The mod \(p\) cohomology of the substack \(\mathcal M_{g,n}\) of smooth pointed curves was computed by S. Galatius in [Topology 43, No. 5, 1105–1132 (2004; Zbl 1074.57013)] in the Harer-Ivanov stable range, identifying a large number of torsion classes.
The main tool used in the paper is the Pontrjagin-Thom collapse map, in particular a generalized version of it holding for differentiable local quotient stacks; this generalization is discussed by the authors in Appendix A. This technique is then applied to the clutching (or gluing) morphisms given by identifying two marked points to form a node and whose images cover the irreducible components of the boundary \(\overline{\mathcal M}_{g,n}\setminus \mathcal M_{g,n}\) of \(\overline{\mathcal M}_{g,n}\). The outcome of this version of Pontrjagin-Thom’s construction is the production of a map from “the homotopy type of the stack” \(\overline{\mathcal M}_{g,n}\), which is a space with the same topological invariants as the stack, to a certain infinite loop space, whose homology is well known (and discussed in Appendix B of the paper). The main result of the paper states the surjectivity of those maps in certain stable ranges. Using this theorem the authors then use the injectivity of the pullback map to detect classes in \(\overline{\mathcal M}_{g,n}\). In particular, large families of torsion classes, which are not reduction of rationally nontrivial classes nor coming from \(\mathcal M_{g,n}\), are shown to exist in \(\overline{\mathcal M}_{g,n}\).

MSC:
14H15 Families, moduli of curves (analytic)
14D23 Stacks and moduli problems
14H10 Families, moduli of curves (algebraic)
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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