Problem 4.1 — ( Topological Schottky Problem).

Definitions and mathematical statement

Problem 4.1, "Topological Schottky Problem," asks to determine the homotopy type of the source's space $\mathcal J_g^c$, and use it to compute its ordinary and compactly supported cohomology, \[ H^\bullet(\mathcal J_g^c),\qquad H_c^\bullet(\mathcal J_g^c). \] It additionally asks to compute its intersection homology. The source gives no definitions of $\mathcal J_g^c$, its superscript $c$, coefficient system, or intersection-homology perversity/support convention. In particular the superscript must not be silently taken to mean a chosen closure or compactification.

The source identifies $g=4$ as the first interesting case and states that $\mathcal J_4^c$ is a singular divisor in $\mathfrak h_4$. Conventionally $\mathfrak h_g$ is Siegel upper half-space, \[ \mathfrak h_g=\{Z\in M_g({\mathbb{C}}):Z^T=Z,\ \operatorname{Im}Z>0\}, \] but the precise Jacobian-locus model, marking and topology require source confirmation.

The following context says that knowledge of $\mathcal J_g$, $\mathcal J_g^c$ or $\mathcal Q_g^c$ should help compute the $\sigma$-invariant homology \[ H_\bullet(\mathcal T_g)^{\langle\sigma\rangle}. \] The spaces $\mathcal Q_g^c$ and $\mathcal T_g$, and the involution $\sigma$, are not defined in the extract. It must not be assumed that $\mathcal T_g$ is contractible Teichmüller space simply from the letter used. For a coefficient ring $R$ in which $2$ is invertible, the source gives the eigenspace decomposition \[ H_\bullet(\mathcal T_g;R) =H_\bullet(\mathcal T_g;R)^+\oplus H_\bullet(\mathcal T_g;R)^-, \] where $\sigma$ acts by $+1$ and $-1$, respectively. This decomposition uses the projections $(1\pm\sigma)/2$ and is not asserted in characteristic $2$ or over arbitrary integral coefficients. The proposed connection is a motivation, not an already specified isomorphism from Jacobian-locus cohomology to these eigenspaces.

Short English statement

Determine the topology, ordinary and compactly supported cohomology, and intersection homology of the specified Jacobian-type locus, especially its singular genus-four case. The source also seeks consequences for invariant homology of another space under an involution. The relevant spaces and comparison maps are not defined in this extract.

Sources

• Local UnsolvedMath registry, permanent ID $11000087$, exact source code catalogue entry, Problem 4.1, "Topological Schottky Problem," PDF page 75. No individual proposer or proposed year is supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The extract includes chapter/page furniture "Finiteness and Torelli spaces," page 69, and a footnote 7 marker without footnote text. The volume has 329 numbered items with repeated local numbering distinguished by PDF page.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; the supplied summary explicitly says it cannot verify status. No separate authoritative definitions or computation references are supplied.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.