Problem 4.3 — Determine good bounds for the homological dimension (or the CW-dimension) ofT c g.

Definitions and mathematical statement

Retain the space written $T_g^c$ in the source's compressed notation "T c g." Its definition and the meaning of the superscript $c$ are absent, so the draft does not identify it with a compact core, compact-type Torelli space, or another completion without further evidence.

Problem 4.3. Determine good upper and lower bounds for the homological dimension, or the CW dimension, of this space. In a fixed ordinary-homology convention with coefficients in $R$, one possible invariant is \[ \operatorname{hd}_R(T_g^c)=\sup\{n:H_n(T_g^c;R)\neq0\}. \] A common homotopy-CW convention would instead use \[ \inf\{\dim X:X\text{ is a CW complex homotopy equivalent to }T_g^c\}. \] The excerpt does not fix coefficients, local-coefficient variants, actual versus homotopy CW dimension, or whether these are the intended definitions. It requests useful bounds rather than providing a proposed formula or a required sharp asymptotic order. The genus range and allowed dependence of bounds must also be specified.

The supplied research summary mentions $\operatorname{cd}(\mathcal I_g)=3g-5$, attributing it to Bestvina--Bux--Margalit, and calls the target a compact core. The space and a comparison with a classifying space of $\mathcal I_g$ are not supplied. A group cohomological-dimension result does not automatically determine ordinary homological dimension or CW dimension of this undefined $T_g^c$. The quoted formula and compact-core interpretation remain source-summary claims requiring review, not assumptions used to derive bounds.

Short English statement

Bound the homological or CW dimension of the source's space $T_g^c$. Its definition and dimension convention are missing, and the cited Torelli-group cohomological-dimension result cannot be transferred without a justified comparison.

Sources

• Local UnsolvedMath registry, permanent ID 11000089, source code catalogue entry, Problem 4.3, PDF page 76. Repeated problem numbering is distinguished by this page. No individual proposer is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The edited volume contains 329 explicitly typeset headings across contributed chapters.
• The supplied summary cites Bestvina--Bux--Margalit for $\operatorname{cd}(\mathcal I_g)=3g-5$ and mentions partial bounds on a compact core, without full bibliography, genus hypotheses, or a comparison theorem identifying the target space.
• Supplied status open and OPEN-TRIAGE, with editorial partial-progress language. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.