Question 4.4 — .

Definitions and mathematical statement

For the usual unpointed genus-$g\geq2$ convention, $\mathcal M_g$ is the moduli stack of smooth compact complex curves and $\overline{\mathcal M}_g$ its Deligne--Mumford compactification. Its added points represent connected stable nodal curves of arithmetic genus $g$, with finite automorphism group. All stable dual graphs are allowed; this is not only the compact-type locus.

The question asks whether the source's Theorem 2.1 can be generalized from smooth-curve moduli to this compactification. The supplied summary describes the intended theorem as Madsen--Weiss-type, but the theorem's actual statement, homotopy/cohomology objects, maps, genus stabilization and range are absent. Accordingly the target is an analogue of that specific theorem for $\overline{\mathcal M}_g$, with its precise objects and comparison supplied, rather than an invented infinite-loop-space equivalence. One must also specify stack/orbifold versus coarse-space interpretation and any marking conventions. The record gives no candidate formula or claim that a related model already proves the desired generalization.

Short English statement

Generalize the missing Theorem 2.1 to Deligne--Mumford stable-curve moduli, identifying the intended homotopy-theoretic comparison and boundary conventions.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Question 4.4, supplied PDF page 106. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. Theorem 2.1 is not supplied.
• The supplied summary identifies a Madsen--Weiss-type question and mentions related Ebert--Giansiracusa homotopy models of the stable-curve moduli stack, without complete bibliography entries or an exact resolving theorem.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000130$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports no complete resolution.