Stable Homology Generators for Compactified Moduli Spaces

Definitions and mathematical statement

For $2g-2+n>0$, let $\overline{\mathcal M}_{g,n}$ denote the Deligne--Mumford moduli stack of connected stable $n$-pointed curves of arithmetic genus $g$: the curves are complete and nodal, the markings are distinct smooth points, and the automorphism group preserving the markings is finite. In the usual complex-topological interpretation, write $H_i(\overline{\mathcal M}_{g,n};{\mathbb{Q}})$ for rational homology. The unmarked compactification is $\overline{\mathcal M}_g=\overline{\mathcal M}_{g,0}$, for $g\geq2$.

The source asks for explicit generators of the stable rational homology of this compactified moduli space. Once the intended stabilization system is specified, denote its degree-$i$ stable group by $H_i^{\mathrm{st}}$. The additive version of the target is an explicitly described collection of classes $\{c_{i,\alpha}\}_{\alpha\in I_i}$ satisfying \[ H_i^{\mathrm{st}}=\operatorname{span}_{{\mathbb{Q}}}\{c_{i,\alpha}:\alpha\in I_i\} \qquad\text{for every }i\geq0. \] If "generators" instead refers to an algebra or to operations induced by gluing curves, those operations must be specified as part of the target. A bare vector-space existence assertion is not the requested explicit description.

The extract supplies neither stabilization maps nor a stable range, and does not say whether genus, number of markings, or both vary. Consequently $H_i^{\mathrm{st}}$ cannot be unconditionally identified here with a particular direct limit or eventual homology group. In particular, no genus-stabilization map on the unmarked compactifications is assumed without a construction. The conventional moduli objects above clarify the subject but do not resolve these missing choices.

Short English statement

Describe explicit generators for the stable rational homology of compactified moduli spaces of curves. The source does not specify the stabilization system or whether generation is additive or uses gluing operations, so those choices remain unresolved.

Sources

• Supplied report: Teichmüller Space (Classical and Quantum) (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/26, pp. 1537--1614; proposed year 2006. https://doi.org/10.4171/owr/2006/26.
• Supplied reference: Tosteson, "stability in Deligne--Mumford compactifications", https://doi.org/10.1017/S0010437X21007582.
• Local UnsolvedMath registry, permanent ID 30000509, catalogue entry; created 25 August 2026, updated 16 September 2026. The record labels the statement corrected_verified following comparison with the official report. Its 21 August 2026 literature triage labels the problem partially solved: stable homology and rational generating-function results are reported, but not a complete explicit generating set of the kind requested. This is a supplied dated assessment requiring independent verification.