Question 5.4 — Is the image of the second Morita class in H8(GL(6, Z); Q)) non-trivial?

Definitions and mathematical statement

Question 5.4 asks whether the image of the second Morita class in the source's degree-$8$ rational group for $\mathrm{GL}(6,{\mathbb{Z}})$ is nonzero. The Morita class, source group and inducing map are not defined in the extract, and the flattened notation $H8$ does not preserve a subscript/superscript distinction.

A natural conventional interpretation uses the abelianization homomorphism \[ a:\operatorname{Out}(F_6)\longrightarrow\mathrm{GL}(6,{\mathbb{Z}}), \] induced from the action on $F_6/[F_6,F_6]\cong{\mathbb{Z}}^6$. If the source's second Morita class is the corresponding $\mu_2\in H_8(\operatorname{Out}(F_6);{\mathbb{Q}})$, the question is \[ a_*(\mu_2)\ne0\quad\text{in }H_8(\mathrm{GL}(6,{\mathbb{Z}});{\mathbb{Q}})\ ? \] This homological reading is explicitly conditional, not a recovered construction of $\mu_2$. Reading the target as cohomology $H^8$ would require a different specified map, because a group homomorphism induces cohomology pullback in the opposite direction. The exact class and variance must therefore be checked against the source. Rank $6$, degree $8$ and rational coefficients are fixed; no stable-rank or integral substitute is made. The source points to [81] for further discussion of $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$ (co)homology but supplies no definition or computation here.

Short English statement

Is the specified second Morita class nonzero after passage to $\mathrm{GL}(6,{\mathbb{Z}})$ in rational degree eight? The natural homological image is displayed conditionally, since the class definition, map and lost homology/cohomology grading need source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $11000296$, exact source code catalogue entry, Question 5.4, PDF page 337. 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 [81] for further free-group automorphism/outer-automorphism cohomology discussion; full entry absent. The second Morita class has no separate construction reference in the record.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and calls for a current primary-literature audit. Supplied OPEN-TRIAGE summary mentions Morita classes and stable/unstable linear-group cohomology but reports no verified decisive computation.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.