On the homotopy type of the Deligne–Mumford compactification

Ebert, Johannes¹ ; Giansiracusa, Jeffrey²

¹Mathematisches Institut, der Universität Bonn, Beringstrasse 1, 53115 Bonn, Germany
²Mathematical Institute, University of Oxford, 24–29 St. Giles’, Oxford, OX1 3LB, United Kingdom

Algebraic and Geometric Topology, Tome 8 (2008) no. 4, pp. 2049-2062

Résumé

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne–Mumford compactification. We give an integral refinement: the classifying space of the Charney–Lee category actually has the same homotopy type as the moduli stack of stable curves, and the étale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney–Lee category.

Détail

DOI : 10.2140/agt.2008.8.2049

Keywords: Deligne–Mumford compactification, moduli of curves, stack, mapping class group, orbit category

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