Problem 1 — Develop techniques to describe the sets Cχ,θ.

Definitions and mathematical statement

Problem 1 asks to develop techniques describing the sets $C_{\chi,\theta}$ defined in the preceding source discussion. The supplied record contains no definition of $C_{\chi,\theta}$, the parameters $\chi,\theta$, their admissible values, the ambient set, membership conditions, or any equivalence relation used in forming these sets.

Thus the precise retained request is to obtain usable descriptions of that source-defined family for its intended parameters. Whether a description should mean an algorithm, generators and relations, a classification, representatives, or an invariant is not specified. No particular criterion or quantifier can be supplied without the missing setup.

The registry places the question in Auroux's discussion of Lefschetz pencils and fibrations and says that the sets encode monodromy of genus-$2$ or genus-$3$ fibrations. This is supplied contextual guidance, not enough to define the sets or to identify $\chi$ as an Euler characteristic or $\theta$ as a chosen monodromy map. Those interpretations must not be guessed from the parameter names.

Short English statement

Develop methods to describe the source's sets $C_{\chi,\theta}$. Their definitions and parameters are missing, so a concrete membership or classification problem cannot yet be specified.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000179$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 1, PDF page 176: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• The registry identifies Auroux's Lefschetz-pencil/fibration chapter as context. The preceding definitions and any additional authoritative citation are absent.