Problem 4.5 — Prove (or disprove) that the natural homomorphism H 4(OutF4; Q)∼= Q−→H 4(IOut4; Q)GL is an isomorphism where the righ…

Definitions and mathematical statement

Let $\operatorname{IOut}_n=\ker(\operatorname{Out}(F_n)\to\operatorname{GL}_n({\mathbb{Z}}))$ in the customary convention. The quotient acts on its rational cohomology by conjugation, well defined because inner conjugations act trivially on cohomology; write $(-)^{\operatorname{GL}}$ for the invariant part under this action.

Problem 4.5 asks to prove or disprove that restriction is an isomorphism \[ r:H^4(\operatorname{Out}(F_4);{\mathbb{Q}})\cong{\mathbb{Q}} \longrightarrow H^4(\operatorname{IOut}_4;{\mathbb{Q}})^{\operatorname{GL}}, \] with the right-hand side generated by a nonzero multiple of the source class $\tau_2$. Equivalently, in this interpretation it asks that the target be one-dimensional and that a generator $u$ of the source restrict to $r(u)=c\tau_2$ for some $c\in{\mathbb{Q}}\setminus\{0\}$. The definition and normalization of $\tau_2$ are absent. The extracted degree placement in "H 4" is ambiguous, but the natural arrow to subgroup invariants is the cohomological restriction convention; ordinary homology has the opposite functorial direction.

The following passage sets up another specific problem without supplying its final question. It reports \[ H^8(M_{3,*};{\mathbb{Q}})\cong{\mathbb{Q}}^2,\qquad H^8(\operatorname{GL}_6({\mathbb{Z}});{\mathbb{Q}})\cong{\mathbb{Q}},\qquad H^8(\operatorname{Out}(F_6);{\mathbb{Q}})\cong{\mathbb{Q}}, \] and the natural maps (9) \[ M_{3,*}\xrightarrow{\ i\ }\operatorname{Out}(F_6) \xrightarrow{\ p\ }\operatorname{GL}_6({\mathbb{Z}}), \] where $i$ is described as injective and $p$ is the homological projection. The meaning of $M_{3,*}$, the construction of $i$ and the degree-index conventions are not restated; a once-marked genus-three mapping class group is the usual possible interpretation, not a supplied definition. No rank, nonvanishing or isomorphism assertion about the induced degree-eight maps is invented.

Short English statement

Is degree-four restriction from $\operatorname{Out}(F_4)$ onto the general-linear invariants of its Torelli subgroup an isomorphism, generated by the named $\tau_2$ class? The class and notation conventions need confirmation.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000312$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 4.5, PDF page 366: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; page-disambiguated numbering among 329 headings.
• Source attributions for the degree-eight groups: Looijenga [61]; Elbaz-Vincent, Gangl and Soulé [13]; Ohashi [91]. Full entries, the definition of $\tau_2$ and the following specific question are absent. The registry cannot verify the modern status.