Problem 4.6 — Determine the homomorphisms H 8(M3,∗; Q) i∗ ←−H 8(OutF6; Q) p∗ ←−H 8(GL(6, Z); Q) (10) induced by the above homomorph…

Definitions and mathematical statement

The preceding local source item 11000312 supplies the group maps omitted from the present excerpt: \[ \mathcal M_{3,*}\xrightarrow{i}\operatorname{Out}(F_6) \xrightarrow{p}\mathrm{GL}_6({\mathbb{Z}}). \] Here $\mathcal M_{3,*}$ is the one-marked-point genus-three mapping class group, $i$ is the natural punctured-surface outer-action injection, and $p$ is induced by abelianization of the rank-six free group. The precise marking conventions must agree with the source. Throughout, $H^8$ means rational group cohomology, so the induced maps reverse direction.

Problem 4.6. Determine \[ H^8(\mathcal M_{3,*};{\mathbb{Q}}) \xleftarrow{i^*}H^8(\operatorname{Out}(F_6);{\mathbb{Q}}) \xleftarrow{p^*}H^8(\mathrm{GL}_6({\mathbb{Z}});{\mathbb{Q}}). \] The adjacent source text reports the vector-space dimensions \[ H^8(\mathcal M_{3,*};{\mathbb{Q}})\cong{\mathbb{Q}}^2,\qquad H^8(\operatorname{Out}(F_6);{\mathbb{Q}})\cong{\mathbb{Q}},\qquad H^8(\mathrm{GL}_6({\mathbb{Z}});{\mathbb{Q}})\cong{\mathbb{Q}}. \] Thus the request includes determining whether $p^*$ and $i^*$ vanish and, if $i^*$ is nonzero, identifying its image line in the two-dimensional pointed mapping-class cohomology. More explicit scalar or coordinate formulas require choices and normalizations of generators/bases; the source does not provide those. The dimensions alone do not determine either homomorphism. No stable-range or rational-vanishing statement is substituted for this specific unstable degree-eight problem.

Short English statement

Compute the two degree-eight rational cohomology pullbacks from $\mathrm{GL}_6({\mathbb{Z}})$ through $\operatorname{Out}(F_6)$ to the pointed genus-three mapping class group, including their vanishing and image.

Sources

• Local UnsolvedMath registry, permanent ID 11000313, source code catalogue entry; no separate proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, chapter "Cohomological structure of the mapping class group and beyond," Problem 4.6, PDF page 366; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; the inventory distinguishes 329 labeled headings by source page.
• Adjacent local ID 11000312 reproduces equation (9) and attributes the three cohomology calculations to Looijenga [61], Elbaz-Vincent--Gangl--Soulé [13], and Ohashi [91], respectively. Full bibliography entries and named generators are not supplied.