Problem 10.3 — Give examples of odd valent graphs Γ whose associated homology classes Φ(Γ)∈ H∗(OutFn; Q) are non-trivial as many as…

Definitions and mathematical statement

Let $\Gamma$ be an odd-valent graph, meaning that every vertex has odd valence, in the source's admissible graph class. Write $\Phi(\Gamma)\in H_d(\operatorname{Out}(F_n);{\mathbb{Q}})$ for its source-defined homology class. The construction of $\Phi$, graph orientation/decoration and degeneracy conventions, and the rules determining $d,n$ are absent.

Problem 10.3 asks for as many examples as possible with $\Phi(\Gamma)\ne0$, and asks to compare these classes with Conant--Vogtmann's explicit cycles in the moduli space of graphs [10]. It does not assert that every odd-valent graph gives a nonzero class.

In the customary kernel notation, let $\operatorname{IOut}_n=\ker p$ for the homological projection $p:\operatorname{Out}(F_n)\to\operatorname{GL}_n({\mathbb{Z}})$, and let $j:\operatorname{IOut}_n\hookrightarrow\operatorname{Out}(F_n)$. The further questions ask, for the constructed classes $\alpha=\Phi(\Gamma)$, whether \[ p_*(\alpha)\ne0\text{ in }H_d(\operatorname{GL}_n({\mathbb{Z}});{\mathbb{Q}}), \qquad\text{or whether}\qquad \alpha\in\operatorname{im}\left(j_*:H_d(\operatorname{IOut}_n;{\mathbb{Q}}) \to H_d(\operatorname{Out}(F_n);{\mathbb{Q}})\right), \] or whether neither description holds. For positive degree, $p_*j_*=0$ because $p\circ j$ is trivial, but vanishing of $p_*(\alpha)$ alone does not imply membership in the image of $j_*$. No false exhaustiveness or exactness assertion is added. The excerpt does not prescribe what form of comparison with the graph-moduli cycles is required.

Short English statement

Find many nonzero odd-valent-graph classes in rational homology of $\operatorname{Out}(F_n)$, compare them with Conant--Vogtmann cycles, and determine their images in general-linear homology or their origins in Torelli-subgroup homology.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000322$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 10.3, PDF page 377: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• Conant and Vogtmann [10] for explicit graph-moduli cycles; the full bibliography entry and preceding definition of $\Phi$ are absent. The registry says only "Partial" without a precise additional result.