Problem 3.5 — Study homomorphisms Modg→ Mod(N ) and Mod(N )→ Modg.

Definitions and mathematical statement

Let $S_g$ be the orientable closed genus-$g$ surface and $N$ the closed nonorientable surface of genus $g$, as in the accompanying source discussion. In the usual conventions $\operatorname{Mod}_g$ uses orientation-preserving mapping classes of $S_g$, $\operatorname{Mod}(N)$ uses mapping classes of $N$, and an asterisk denotes the extended orientable mapping class group, allowing orientation reversal. The allowed genus range and any exceptional conventions are not stated.

Problem 3.5. Study homomorphisms in both directions, \[ \operatorname{Mod}_g\longrightarrow\operatorname{Mod}(N), \qquad\operatorname{Mod}(N)\longrightarrow\operatorname{Mod}_g. \] The source does not specify a classification up to conjugacy, a finite-image criterion, injective homomorphisms only, or another narrower target; none is silently imposed.

The source credits Birman and Chillingworth [3] with relating $\operatorname{Mod}(N)$ to a subgroup of $\operatorname{Mod}_{g-1}^*$, modulo an element of order two, through the orientation double cover $S_{g-1}\to N$. It then states that certain finite-index subgroups of $\operatorname{Mod}(N)$ are isomorphic to subgroups of $\operatorname{Mod}_{g-1}$. These are supplied structural context. The exact lift subgroup and quotient in the first statement are not defined: the deck involution must be normal in the subgroup being quotiented, and a quotient of the entire extended mapping class group by an arbitrary order-two subgroup is not thereby defined. The draft does not replace the finite-index embedding statement by an embedding of the full $\operatorname{Mod}(N)$, or confuse the cover's genus $g-1$ with the genus $g$ groups in the two requested directions.

Short English statement

Investigate homomorphisms between the orientable and nonorientable mapping class groups in both directions. The orientation-double-cover comparison supplies context involving genus $g-1$ and finite-index subgroups, but does not itself classify those homomorphisms.

Sources

• Local UnsolvedMath registry, permanent ID 11000123, source code catalogue entry, Problem 3.5, PDF page 98. The chapter header identifies M. Korkmaz; no individual proposer field is supplied.
• 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 volume contains 329 explicitly typeset headings across chapters, with repeated local numbering disambiguated by page.
• Birman and Chillingworth [3] is cited for the orientation-double-cover comparison. The full bibliographic entry, lift subgroup, exact quotient, and low-genus hypotheses are not provided.
• Supplied status open and OPEN-TRIAGE. The summary mentions studied rigidity/finiteness results and says the problem is not fully settled, without precise references. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.