Problem 8 — [Contributed by the referee] Does Theorem 5 say anything about the structure of the end of Riemann’s moduli space?

Definitions and mathematical statement

The referee-contributed question asks what the source's Theorem 5 implies about the end of Riemann's moduli space, especially homology near infinity. For a fixed compact exhaustion $K_a$ of the intended moduli space $\mathcal M(F)$, one possible precise end object is the inverse system $\{H_j(\mathcal M(F)\setminus K_a)\}_a$; the source does not choose this system, its limit or coefficients. Theorem 5 is absent, so no consequence of it is invented.

The following proposed mechanism uses $F=F_{g,r}^s$, with $s\geq0$ punctures, and a space $F^\times$ called its one-point compactification, identifying all punctures. The source says $F^\times=F$ iff $s\leq1$; with punctures interpreted as deleted ends, this is not literal equality after one-point compactification, so the marked-point/deleted-end convention must be checked. Let $C_p(\operatorname{Arc}(F))$ be the source's arc cell complex with boundary $\partial$. Neither the quotient/cell conventions nor coefficients are provided.

A cell $\sigma[\alpha]$ comes from an arc family $\alpha$, and a codimension-one face $\sigma[\beta]$ removes one arc. The source separates these faces according to whether \[\operatorname{rank}H_1(F^\times\setminus\!\bigcup\beta) \quad\text{equals or differs by one from}\quad \operatorname{rank}H_1(F^\times\setminus\!\bigcup\alpha).\] The two contributions define $\partial=\partial_1+\partial_2$, with $\partial_2$ in the rank-changing case, and are reported to satisfy \[\partial_1^2=\partial_2^2=0,\qquad \partial_1\partial_2+\partial_2\partial_1=0.\] The source's bicomplex has chains on $p$-cells in bidegree \[v=-\operatorname{rank}H_1(F_\alpha),\qquad u=p-v,\] with $\partial_1$ of bidegree $(-1,0)$ and $\partial_2$ of bidegree $(0,-1)$. The cut space $F_\alpha$ is not defined here or explicitly identified with the complement above. Its stated spectral sequence has initial differential $\partial_2$ and converges to $H_*(\operatorname{Arc}(F))$, with convergence/filtration conventions requiring the full construction.

The text explicitly says the following argument is complicated and has not been independently checked, and calls it neither a settled problem nor a theorem: it believes the spectral sequence collapses "in its $E^1$-term to its top horizontal row except in dimension zero." It then claims the homology of $\operatorname{Arc}$ is the $\partial_1$-homology of the $\partial_2$-kernels in that row. These words could mean concentration at $E^1$, followed by horizontal homology, rather than literal $E^1=E^\infty$; the page/degeneration and zero-dimensional exception must be specified. The kernel row must not be replaced by the entire row. Separately it reports [31] that $\partial_1$-homology of the top row itself agrees with that of uncompactified $\mathcal M(F)$.

Finally, the source reports [30,46] that stratifications of arc complexes for bordered surfaces induce a stratification of $A(F)$ for punctured $F$, and suggests revisiting calculations [19] and [16,22] while avoiding technical difficulties of the Deligne--Mumford compactification $\overline{\mathcal M}(F)$. The space $A(F)$, those calculations and the proposed comparison are not supplied; this is a tentative application, not an established end-homology computation.

Short English statement

Determine what the omitted theorem says about homology near the end of moduli space. The proposed arc-bicomplex mechanism and its top-row spectral-sequence claim are preserved with the source's explicit unchecked-argument warning.

Sources

• Local UnsolvedMath registry, permanent ID $11000138$, exact source code catalogue entry.
• Anonymous referee, contribution in Farb, Problems on Mapping Class Groups and Related Topics, Problem 8, PDF page 119: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• Theorem 5; [31] for top-row/moduli homology; [30,46] for stratifications; and [19], [16,22] for proposed calculations are invoked without reproduced statements or full bibliography.