Problem 7.2 — Find explicit graphs Γ∈G odd such that the corresponding homology classes Φ(Γ) are non-trivial.

Definitions and mathematical statement

Let $\mathcal G_{\mathrm{odd}}$ and $\Phi$ be the graph family and homology-class construction introduced before this problem. The task is to exhibit explicit graphs \[\Gamma\in\mathcal G_{\mathrm{odd}}\qquad\text{such that}\qquad\Phi(\Gamma)\ne0.\] The graph conditions, orientations/decorations, construction, target groups, ranks/degrees and coefficients are absent from the supplied excerpt. The research summary suggests odd-valent graphs and free-group outer-automorphism homology, but that does not identify all admissibility conditions or the actual map. In particular it does not justify replacing $\mathcal G_{\mathrm{odd}}$ by all finite odd-valent graphs or changing the source's homology target to cohomology.

An explicit example should specify the finite graph together with every datum required by the definition, and prove that the associated class is nonzero in its stated target. If $\Phi(\Gamma)$ is represented by a cycle $z_\Gamma$ in a chain complex, the required assertion is that $z_\Gamma$ is not a boundary, not merely that the graph or chain itself is nonzero. A nonzero pairing with an appropriate cohomology class would be one possible certificate, but no method is prescribed. The source does not specify the number of examples, require an infinite family, or provide the positive-degree/rank restrictions needed to select a candidate.

Short English statement

Exhibit concrete admissible graphs whose associated homology classes are nonzero. The family and class map are undefined in the excerpt, so an odd-valent graph alone is not a certified example.

Sources

• Local UnsolvedMath registry, permanent ID $11000319$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 7.2, PDF page 372: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• The summary mentions graph complexes and Conant--Vogtmann-related work without a precise citation or stated nonvanishing theorem.