Problem 1.1 — Study the function Ψ: Mg→ [−log(µg),∞) as a (rational) Morse function.

Definitions and mathematical statement

For a genus $g$ in the source's range, let $\mathcal M_g$ denote its moduli space of genus-$g$ curves/Riemann surfaces and let \[\Psi:\mathcal M_g\longrightarrow[-\log(\mu_g),\infty)\] be the particular function introduced in the omitted preceding text. The parameter $\mu_g$ must be positive for the stated real logarithm. The codomain states a lower bound and does not by itself assert that the lower bound is attained or that $\Psi$ is surjective.

The programme has three tasks: study this $\Psi$ as a rational Morse function; classify its rational critical points; and derive properties of $H^*(\mathcal M_g;{\mathbb{Q}})$, including the specific statements referred to later in the source. Neither the formula for $\Psi$, $\mu_g$, the meaning of rational Morse/critical point nor the later statements are supplied. In particular "rational critical point" cannot silently be interpreted as an algebraic point defined over ${\mathbb{Q}}$, or as an ordinary smooth critical point, without that definition.

If the intended framework is ordinary orbifold Morse theory, one would analyze local lifts of $\Psi$, their critical loci, Hessian indices and isotropy, and explain the resulting filtration/attachments computing rational cohomology. This describes one conditional framework, not an assertion that the given function is smooth, proper or Morse, nor a replacement of the source's rational notion. The moduli space has finite-automorphism/orbifold issues and is noncompact in the usual closed-surface setting, so these features must enter any claimed cohomological deduction. No universal cohomology presentation or stable-range result is added as a substitute for this function-specific programme.

Short English statement

Study the specified moduli-space function through its intended rational Morse theory, classify its critical points and extract rational cohomological consequences. The defining function and rational terminology are missing from this excerpt.

Sources

• Local UnsolvedMath registry, permanent ID $11000001$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 1.1, PDF page 11: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction. The summary attributes this programme to Farb (2006).
• The summary names Harer and Madsen--Weiss for other rational-cohomology/stable-range progress, without exact citations; it does not claim they resolve the specified Morse programme.