Problem 5.23 — Compute H1(Modg[L]; Z).

Definitions and mathematical statement

In the usual closed oriented genus-$g$ surface convention, the level-$L$ subgroup is \[ \operatorname{Mod}_g[L] =\ker\left(\operatorname{Mod}_g\longrightarrow \operatorname{Sp}(H_1(\Sigma_g;{\mathbb{Z}}/L{\mathbb{Z}}))\right), \] for an allowed positive integral level $L$. The problem is to compute its integral first homology, equivalently its abelianization, \[ H_1(\operatorname{Mod}_g[L];{\mathbb{Z}}) \cong\operatorname{Mod}_g[L]/[\operatorname{Mod}_g[L],\operatorname{Mod}_g[L]], \] with an explicit group description, for example its invariant factors. The exact genus and level ranges are not repeated in the problem sentence.

The source reports that for $g\geq3$ this group is finite, proved independently by McCarthy and Hain, referring to [Ha2, Proposition 5.2]. Thus computing it asks for more than establishing finiteness. A footnote says Hain proves a stronger result computing $H^1(\operatorname{Mod}_g[L];V)$ for finite-dimensional symplectic representations $V$; that is a cohomology statement with coefficients and must not be confused with the requested integral homology group.

The continuation says a following conjecture, discussed in [Ha2, Section 5], would imply rank one for the orbifold Picard group of the moduli space of level-$L$ structures, and reports that Picard group's finite generation from the preceding Hain--McCarthy result. The following conjecture is absent. These are reported motivations, not an asserted rank-one consequence of the computation alone.

Short English statement

Compute the full abelianization of the level mapping class subgroup, beyond its reported finiteness for $g\geq3$, retaining the distinction from Hain's coefficient-cohomology result.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 5.23 and its continuation/footnote, supplied PDF page 51. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• McCarthy and Hain, independent finiteness attribution; Hain [Ha2, Proposition 5.2 and Section 5]. The complete bibliography entry is missing. The footnote credits Hain's stronger coefficient-cohomology computation.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000067$, exact source code catalogue entry. Its OPEN-TRIAGE summary mentions a level-$2$ partial result with truncated attribution "Sur-..."; no complete author or citation is supplied.