Nontrivial Actions on Invariant Prym Homology

Definitions and mathematical statement

Use the notation of the supplied Prym-homology setting: $H_K$ denotes the Prym homology under consideration, $\mathcal G_K^\circ$ denotes the identity component of the group acting on it, and $\mathcal D_K\trianglelefteq\mathcal G_K^\circ$ is the connected normal subgroup generated by the one-parameter subgroups arising from lifted Dehn twists. Define the invariant part by \[ H_K^{\mathcal D_K}=\{v\in H_K: d\cdot v=v\text{ for every }d\in\mathcal D_K\}. \] Normality makes this invariant part stable under $\mathcal G_K^\circ$, and the action factors through $\mathcal G_K^\circ/\mathcal D_K$, with \[ (g\mathcal D_K)\cdot v=g\cdot v. \] Question. Does there exist an example in the intended Prym-homology setting such that \[ H_K^{\mathcal D_K}\neq\{0\} \quad\text{and}\quad \exists\,g\in\mathcal G_K^\circ\;\exists\,v\in H_K^{\mathcal D_K} \text{ with }g\cdot v\neq v? \] Thus the requested quotient action must be nontrivial, not merely defined on a nonzero invariant part.

Unresolved qualifications. The supplied record does not define the admissible data $K$, the underlying surface or covering, the construction or coefficient domain of $H_K$, or the construction and category of $\mathcal G_K$. It also does not specify which lifted Dehn twists yield the one-parameter subgroups, or the sense of connectedness and subgroup generation. These definitions must be recovered from the cited report before the existential question has a fully specified domain. The display above records the intended question conditionally on that missing setting.

Short English statement

The question asks for a Prym-homology example whose subspace fixed by all the lifted-Dehn-twist one-parameter subgroups is nonzero and still carries a nontrivial action of $\mathcal G_K^\circ/\mathcal D_K$. The supplied record leaves the precise class of admissible examples unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002908$, title Nontrivial Actions on Invariant Prym Homology, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: New Perspectives on the Interplay between Discrete Groups in Low-Dimensional Topology and Arithmetic Lattices (2016), Oberwolfach Reports, pp. 1701--1746, DOI 10.4171/owr/2015/30. Supplied source and literature-review link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2015/30. The registry describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked on 2026-08-21: status "open"; the extracted request for examples with nonzero homology and nontrivial action is described as unresolved in the cited record, with no verified complete answer located. This is supplied dated curation derived from the OWR record, not an independently verified assessment here. Time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that the official report passage was checked directly and that the question, definitions, and notation were restored without strengthening the claim. These are supplied verification claims; this draft does not claim to have visited or checked the report.