Problem 3.4 — Prove that all the secondary classes d2,d 3,··· are non-trivial.

Definitions and mathematical statement

Problem 3.4 asks to prove that every secondary class \[ d_i\quad(i\ge2) \] is nonzero in its source-defined target. The construction, degrees, coefficients, groups/spaces and genus range of these classes are absent from the extract; nonvanishing cannot be assigned to an invented cohomology group or stable range.

The context then asks about the first class $d_1$, viewed there as a numerical invariant on $K_g$. Conventionally $K_g$ is the Johnson kernel, but its precise definition and $d_1$ normalization need source confirmation. If a separating simple closed curve $C\subset\Sigma_g$ cuts the closed surface into genera $h$ and $g-h$, the source reports, for its Dehn twist $\tau_C\in K_g$, \[ d_1(\tau_C)=c_g\,h(g-h) \] under the usual multiplicative reading of "up to a constant depending on $g$." The value of $c_g$, twist direction and coefficient convention are not supplied and are not set to one by default.

The further task is to give an algorithm evaluating $d_1(\phi)$ for a specified $\phi\in K_g$, for example by analyzing its action on $\pi_1(\Sigma_g)$. The representation of $\phi$, based versus outer action, and output normalization must be fixed. The source's statement that no such algorithm is currently known is its historical assessment, not a present-day verification. The special separating-twist formula is not asserted to supply a general algorithm or to prove nontriviality of the higher secondary classes.

Short English statement

Prove nonvanishing of all the specified secondary classes beyond $d_1$, and separately compute $d_1$ effectively on arbitrary elements of the source's Johnson kernel. The known separating-twist formula retains its unspecified genus-dependent normalization.

Sources

• Local UnsolvedMath registry, permanent ID $11000307$, exact source code catalogue entry, Problem 3.4, PDF page 363. 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.
• 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 a Torelli/Morita--Hain secondary-class framework and partial nontriviality, but gives no precise cases or complete references. No separate defining citation for $d_i$ is supplied.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.