Problem 4.9 — Investigate the topology of Hg andHc g and their components.

Definitions and mathematical statement

Retain the source's spaces $\mathcal H_g$, $\mathcal H_g^c$, their components $\mathcal H_{g,\alpha}^c$, the target $\mathfrak h_g$, and $\mathcal J_g^c$, and its group notation $T_g$. Their definitions are not included in the excerpt, so no identification with similarly named spaces or Lie algebras is presumed.

Problem 4.9. Investigate the topology of $\mathcal H_g$, $\mathcal H_g^c$, and their components. Specifically, compute their homology and the cohomology at infinity of each $\mathcal H_{g,\alpha}^c$. With a coefficient system $R$ fixed, a usual ordinary-cohomology-at-infinity convention is \[ H^j_\infty(X;R)=\varinjlim_{K\subset X\ \mathrm{compact}}H^j(X\setminus K;R), \] with restriction maps as $K$ increases. This is distinct from compactly supported cohomology. The source does not select coefficients or a convention for infinity, nor state the genus range or component indexing.

The source says the period map immerses each $\mathcal H_{g,\alpha}^c$ in $\mathfrak h_g$ as a closed subvariety; it concludes that each is a Stein manifold of complex dimension $2g-1$, and hence has the homotopy type of a CW complex of that dimension. Those are retained as supplied structural claims, with definitions and the required smoothness/closed-map properties missing. Under the stated Stein-manifold hypothesis the usual CW consequence supplies an upper bound \[ \operatorname{hdim}(\mathcal H_{g,\alpha}^c)\leq2g-1, \] not by itself a lower bound or an assertion that every minimal CW model has dimension exactly $2g-1$. The immersion/closed-subvariety wording must be checked before treating its Stein conclusion as established.

The adjacent numbered note says that the number of components of $\mathcal H_g$ grows exponentially with genus while the minimum number of generators of $T_g$ grows polynomially. It says the failure of an earlier genus-three argument in higher genus suggests that $\mathcal J_g^c$ is generally not simply connected. The earlier argument, group definition, rates, and hypotheses are absent, and this suggestion is not converted into a proved non-simple-connectivity assertion.

Short English statement

Compute the homology of the source's $\mathcal H_g$, $\mathcal H_g^c$, and components, together with cohomology at infinity of $\mathcal H_{g,\alpha}^c$. Its Stein and component-growth discussion is retained conditionally; the defining spaces and coefficient conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID 11000094, source code catalogue entry, Problem 4.9, PDF page 79. No individual proposer is supplied. Page identifies the item despite repeated chapter numbering.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The edited volume contains 329 explicitly typeset headings across contributed chapters.
• The statement supplies the period-map/Stein/CW discussion and a detached note numbered 10 about component growth and a genus-three argument, but no separate reference or earlier definitions.
• Supplied status open and OPEN-TRIAGE. The research summary calls it a research program and says a settled answer cannot be verified. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.