Problem 12.1 — Prove that the above characteristic classes induce surjective homomorphism H3(BDiffδ +Σg; Z)−→R2 for any g.

Definitions and mathematical statement

Let $\operatorname{Diff}^{\delta}_+(\Sigma_g)$ denote the source's orientation-preserving diffeomorphism group with the discrete topology, and $B$ its classifying space. Problem 12.1 asks to prove that the preceding characteristic classes induce a surjective homomorphism \[ H_3(B\operatorname{Diff}^{\delta}_+(\Sigma_g);{\mathbb{Z}}) \longrightarrow{\mathbb{R}}^2 \] for every genus in the source's intended range. The defining classes and their normalizations are absent, so the homomorphism itself is not yet identified.

If the intended pair is $\alpha_g,\beta_g\in H^3(B\operatorname{Diff}^{\delta}_+(\Sigma_g);{\mathbb{R}})$, its evaluation map would be \[ z\longmapsto(\langle\alpha_g,z\rangle,\langle\beta_g,z\rangle). \] Surjectivity means that every pair of real numbers is attained by an integral homology class. It is not merely nonvanishing, independence of the cohomology classes, or surjectivity after replacing the domain by real homology. This display is a conditional explanation of the omitted characteristic map, not a choice of new classes.

The context says the classes in (22) are stable under genus change. It reports [57,58] relating twisted mapping-class-group cohomology to flux and Calabi homomorphisms, and using this to construct nontrivial classes of $B\operatorname{Symp}^{\delta}\Sigma_g$. The symplectic form, coefficient system, exact constructions and stabilization maps are missing. It says all then-known classes of both classifying spaces were stable, and introduces a following problem not provided. That observation is not asserted to apply to every possible class or to the present literature.

Short English statement

Prove full surjectivity onto the real plane of the specified characteristic-number map from third integral homology of the discrete diffeomorphism group, in every intended genus. The classes defining the map are missing. Stability and symplectic-topology remarks remain separate source context.

Sources

• Local UnsolvedMath registry, permanent ID $11000327$, exact source code catalogue entry, Problem 12.1, PDF page 380. No individual proposer or proposed year is supplied; the title is truncated.
• 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 formula (22); references [57,58] for twisted-cohomology/flux/Calabi interactions and constructed symplectomorphism classes; [68] for symplectic-topology background. Full entries, the classes above and the following problem are absent.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied OPEN-TRIAGE summary mentions surface-bundle bordism/Morita context and partial surjectivity knowledge but explicitly cannot verify the question or supply precise results.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.