Problem 2.2 — Letγ1,···,γ 2g be the standard basis of Z2g.

Definitions and mathematical statement

Let $\gamma_1,\ldots,\gamma_{2g}$ be the standard basis of the free abelian group ${\mathbb{Z}}^{2g}$, and retain the source's space $Y_g$ and functions $L_\gamma$. Define \[ L=\sum_{i=1}^{2g}L_{\gamma_i}:Y_g\longrightarrow[0,\infty). \] The source excerpt does not define $Y_g$, $L_\gamma$, or the genus range. The research summary calls $Y_g$ Torelli space. In the usual closed oriented surface interpretation this would mean Teichmüller space modulo the Torelli group, the kernel of the mapping-class-group action on integral first homology, but that setup must be checked against the source.

Problem. Study $L$ as a Morse function: find its critical sets and use them to deduce properties of the topology of $Y_g$. In particular, give upper and lower estimates for its homotopy dimension. In the usual CW-type convention, \[ \operatorname{hdim}(Y_g) =\inf\{\dim X:X\text{ is a CW complex homotopy equivalent to }Y_g\}. \] The extracted phrase "homotopy dimension 2 of $Y_g$" contains a detached $2$, plausibly a footnote marker. It is not treated as a demand to prove that $Y_g$ has dimension $2$; its role requires source confirmation.

A rigorous Morse-theoretic formulation must establish the smooth or stratified setting, regularity of $L$, and the relevant nondegeneracy and global hypotheses. The request for "critical sets" does not by itself specify ordinary Morse, Morse--Bott, or another version of Morse theory. Without the definitions of the summands, no differential or explicit critical-point condition can be supplied.

Short English statement

Analyze the sum of the source's $2g$ basis-indexed functions on $Y_g$ by Morse theory, determine its critical sets, and use them to bound the homotopy dimension. The space and summands are omitted, and a detached $2$ may be a footnote marker.

Sources

• Local UnsolvedMath registry, permanent ID 11000003, source code catalogue entry, Problem 2.2, PDF page 13. No individual proposer is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 2.2 at the specified page. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The edited volume has 329 explicitly typeset Problem, Conjecture, and Question headings across chapters, with repeated local numbers disambiguated by PDF page.
• The research summary attributes the program to Farb's chapter (2006), identifies the space as Torelli space, mentions later topology and homological-dimension work without citations, and says no dedicated published Morse treatment of this particular $L$ was located. These are supplied assessments.
• Supplied status open and OPEN-TRIAGE. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.