Problem 3.5 — Find explicit way of calculating d1(ϕ) for any given element ϕ ∈ Kg.

Definitions and mathematical statement

Let $\mathcal K_g$ be the source's Johnson kernel and let $d_1$ be its previously defined class/homomorphism. The main task is to give an explicit method calculating $d_1(\varphi)$ for every $\varphi\in\mathcal K_g$. Its definition, normalization, codomain and input encoding are absent, so no numerical formula for a separating twist or other element is fabricated. A word-input formulation must specify generators and promise membership in $\mathcal K_g$.

The first concrete question concerns the Magnus representation \[r:\mathcal I_{g,1}\to GL(2g;{\mathbb{Z}}[H]),\] where $H$ is the relevant first-homology module and ${\mathbb{Z}}[H]$ its group ring. Does it detect $d_1$? Once the domain/lift of $\mathcal K_g$ is identified, detection means that equal representation values give equal $d_1$-values; for a homomorphism $d_1$ this is equivalent to \[\ker r\cap\mathcal K_g\subseteq\ker d_1,\] or factorization of $d_1$ through $r(\mathcal K_g)$. The source cites Suzuki [103] for nonfaithfulness of $r$, but explicitly only suggests that this inclusion might fail. Nonfaithfulness alone does not decide it. Since the displayed representation is on $\mathcal I_{g,1}$ while $\mathcal K_g$ may use the closed-surface convention, compatible lifts/quotients are essential missing data.

The second concrete question uses Jones's representation [45] of the hyperelliptic mapping class group $\Delta_g$: does its restriction to $\Delta_g\cap\mathcal K_g$ detect $d_1$? With that representation denoted by $J$, the homomorphism reading asks \[\ker J\cap\mathcal K_g\subseteq\ker d_1\] inside the common domain, equivalently whether the restricted $d_1$ factors through its image. Its coefficient/parameter/normalization conventions are not supplied. Kasahara [46] is cited only as related work for $g=2$.

The source also names Endo [14] and Morifuji [71] for hyperelliptic work, Kitano [54], and Hain--Reed [33] as related to $d_1$. It reports a Biss--Farb [5] claim that $\mathcal K_g$ is not finitely generated for all $g\geq3$, Mess [69] for $\mathcal K_2$ as an infinitely generated free group, and says finite generation of its abelianization was still unknown, referring to Problem 2.2 of [80]. The historical Biss--Farb statement must be checked against primary sources and subsequent corrections before being treated as a current theorem; even non-finite-generation alone would not settle abelianization. Finally it reports Kawazumi [50] developing harmonic Magnus expansions giving differential forms representing Mumford--Morita--Miller classes on the universal curve over $\mathcal M_g$. This contextual report is not an explicit solution to the $d_1$ calculation or detection questions.

Short English statement

Compute the specified $d_1$ invariant on every Johnson-kernel element and determine whether Magnus or hyperelliptic Jones representations retain it. The invariant definition, compatible domains and several historical status assertions need checking.

Sources

• Local UnsolvedMath registry, permanent ID $11000308$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 3.5, PDF page 363: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• Suzuki [103]; Jones [45]; Kasahara [46]; Endo [14]; Morifuji [71]; Kitano [54]; Hain--Reed [33]; Biss--Farb [5]; Mess [69]; Problem 2.2 of [80]; and Kawazumi [50]. All are supplied internal references without full bibliographic entries.