Problem 4.3 — Produce non-trivial rational (co)homology classes of OutFn.

Definitions and mathematical statement

Let $F_n$ be free of rank $n$ and $\operatorname{Out}F_n=\operatorname{Aut}(F_n)/\operatorname{Inn}(F_n)$. The principal task is to construct nonzero rational homology or cohomology classes \[ x\in H_k(\operatorname{Out}F_n;{\mathbb{Q}}) \quad\text{or}\quad x\in H^k(\operatorname{Out}F_n;{\mathbb{Q}}), \] with ranks and positive degrees explicitly stated and nonvanishing justified. Positive degree is the substantive interpretation; degree-zero classes are automatic. The source does not require all ranks or a full ring computation.

The continuation defines the homologically trivial outer subgroup $I\operatorname{Out}_n=\ker(\operatorname{Out}F_n\to\operatorname{GL}(n,{\mathbb{Z}}))$. It reports Igusa [38] constructing higher Franz--Reidemeister torsion classes \[ \tau_{2i}\in H^{4i}(I\operatorname{Out}_n;{\mathbb{R}}), \] and relates them to the source's Theorem 4.1: for the natural projection $p$, the Borel classes satisfy \[ p^*\beta_{4i+1}=0\quad\text{in }H^{4i+1}(\operatorname{Out}F_n;{\mathbb{R}}), \qquad\beta_{4i+1}\in H^{4i+1}(\operatorname{GL}(n,{\mathbb{Z}});{\mathbb{R}}). \] The exact rank/degree ranges and construction are omitted. The source says nontriviality of the $\tau_{2i}$ was unknown; vanishing of the pulled-back Borel class alone is not asserted to prove that nontriviality.

The registry marks the item solved, but its summary says only partial classes are known and gives an unfinished $H_2$ rank claim without an exact theorem. These reports do not identify a solution of the separate real-coefficient torsion question.

Short English statement

Construct positive-degree rational classes for outer automorphism groups, while keeping separate the reported real higher-torsion classes on their Torelli kernels and the question of their nonvanishing.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 4.3 and following discussion, supplied PDF page 365. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Igusa [38], higher torsion, and the preceding source Theorem 4.1 for Borel-class pullback vanishing. The bibliography, theorem hypotheses and class definitions are absent.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000310$, exact source code catalogue entry; solved/SOLVED-IN-LITERATURE label with a PARTIAL summary mentioning Morita/MMM-type work and an incomplete $H_2$ assertion, without exact citations.