Problem 13 — What are the kernels of the Magnus representations?

Definitions and mathematical statement

Problem 13 asks to determine the kernels of the source's Magnus representations. For any specified representation $\rho:G\to\operatorname{Aut}(V)$, its kernel is \[ \ker\rho=\{g\in G:\rho(g)=\operatorname{id}_V\}. \] The record does not define which Magnus representations are intended: their domains, module/covering data, coefficient rings and possible semilinear conventions are absent. In a common setting such constructions use an action on homology of an abelian cover; this terminology does not select a unique representation or justify replacing the plural request by a familiar single case.

The following contextual paragraph reports that [48] contracts powers of a canonical $1$-cocycle to obtain combinatorially explicit cycles and cocycles on $\mathcal M(F)$, generating the tautological algebra. That cocycle, its coefficient system, contractions and the surface/moduli conventions are not supplied. It then asks about extending these classes to a screen model for $\overline{\mathcal M}(F)$ and to a combinatorial compactification, and revisiting [16,19,22] in that setting.

For a cohomology class $\alpha\in H^*(\mathcal M(F);R)$, an ordinary extension across an inclusion $i:\mathcal M(F)\hookrightarrow X$ would mean a class $\overline\alpha\in H^*(X;R)$ with $i^*\overline\alpha=\alpha$. This is conditional terminology: the source mentions both cycles and cocycles, does not fix support/coefficient conventions, and does not define either compactification. It supplies no proposed extension formula or link between these extension questions and the Magnus kernels. All are retained without inventing a missing canonical cocycle or compactification.

Short English statement

Determine the kernels of the specified Magnus representations. The extract also proposes extending tautological classes from a canonical cocycle to two compactification models. The representations, cocycle and models are undefined here, so the main and contextual requests remain separate source dependencies.

Sources

• Local UnsolvedMath registry, permanent ID $11000143$, exact source code catalogue entry, Problem 13, PDF page 122. No individual proposer or proposed year is supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume has 329 explicitly typeset numbered items across chapters, repeated local numbering distinguished by PDF page.
• Internal reference [48] for contracted canonical-cocycle constructions and tautological generators; references [16,19,22] are proposed for reconsideration in the compactified context. Complete bibliography entries are absent.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied OPEN-TRIAGE summary mentions Magnus-kernel studies for pure braid/surface groups and containment in derived subgroups, but supplies no precise representation, theorem hypotheses or citations; its general not-fully-computed assessment is unverified here.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.