Problem 2.3 — Work out the details of this construction of the completion Yg of Yg.

Definitions and mathematical statement

The extraction has lost the decorations distinguishing a space from its completion. Write $Y_g$ for the original space and $\overline Y_g$ for the proposed completion, solely to distinguish the two roles. The preceding definitions of these spaces, their construction, the allowed genera and the function $L$ are not supplied.

The problem asks to complete that construction and prove that \[ L:Y_g\longrightarrow[0,\infty) \quad\text{extends to}\quad \overline L:\overline Y_g\longrightarrow[0,\infty) \] as a proper map, meaning continuous with compact inverse image of every compact set. Ideally the inclusion $Y_g\hookrightarrow\overline Y_g$ should be a homotopy equivalence, and $\overline Y_g\setminus Y_g$ should be the boundary of a manifold-with-corners structure on $\overline Y_g$. These latter goals are explicitly aspirational in the source.

Study $\overline L$ as a "Morse" function, allowing nonisolated critical points, and explain the change in homotopy type of $\overline L^{-1}([0,t])$ when $t$ passes through a critical value. The required smooth/stratified structure and meaning of critical point are not defined in the excerpt.

For $g=2$, the source describes added pieces with extracted label $Y_1^1\times Y_1^1\times S^1$, identified there with $\mathbb H^2\times\mathbb H^2\times S^1$. The restricted function factors through the first two factors, with a unique minimum and no other critical points, hence a circle of minima. Its stated hope is that a point of $\overline Y_2$ is a local minimum exactly when it is in one of these minimum circles, and that additional isolated index-$1$ critical points connect those circles in a tree-like pattern. This would give $\overline Y_2$ and $Y_2$ the homotopy type of a wedge with one circle for every intended splitting, recovering the cited Mess theorem. The label convention and the indexing splittings are missing, and the proposed critical-point picture is not asserted as proved.

Short English statement

Construct the proposed completion of $Y_g$, extend its length-like function properly, and analyze sublevel-set topology, with a conjectural genus-two minimum-circle and index-one critical-point picture recovering Mess's result.

Sources

• Farb, chapter in Problems on Mapping Class Groups and Related Topics, Problem 2.3, supplied PDF page 13. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. The preceding completion construction and definitions are absent.
• The supplied summary cites Mess (1992) for the theorem that $I_2$ is free of infinite rank, without a full bibliography entry. The source says the hoped-for genus-two analysis should recover the previously mentioned theorem.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000004$, exact source code catalogue entry; supplied OPEN-TRIAGE status and programmatic open report.