On the homotopy type of the Deligne-Mumford compactification

TL;DR

This paper refines a classical result by showing that the classifying space of the Charney-Lee category has the same homotopy type as the moduli stack of stable curves, linking topology and algebraic geometry.

Contribution

It provides an integral and homotopical refinement of Charney and Lee's theorem, connecting the classifying space with the moduli stack of stable curves.

Findings

Classifying space of Charney-Lee category has the same homotopy type as the moduli stack.

Etale homotopy type of the moduli stack is equivalent to the profinite completion.

Refines rational homology equivalence to an integral homotopy equivalence.

Abstract

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 etale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category.