Local Branching of Holonomy Maps for Geometric Structures

Definitions and mathematical statement

A closed manifold is a compact manifold without boundary. In the usual homogeneous-space setup, a geometric model consists of a manifold $X$ and a Lie group $G$ acting transitively on $X$. A $(G,X)$-structure on a manifold $M$ of the same dimension as $X$ is a maximal atlas of charts into $X$ whose transition maps are locally restrictions of elements of $G$. The supplied record does not specify the admissible geometric models, regularity, or further hypotheses on the action.

For connected $M$, the usual developing-map description associates to such a structure a local diffeomorphism $\operatorname{dev}\colon\widetilde M\to X$ and a holonomy homomorphism $\rho\colon\pi_1(M)\to G$ satisfying \[ \operatorname{dev}(\gamma\cdot p)=\rho(\gamma)\cdot\operatorname{dev}(p) \qquad(\gamma\in\pi_1(M),\ p\in\widetilde M). \] The holonomy homomorphism is defined up to conjugation when no additional framing is chosen. The source does not specify whether $M$ must be connected; the description above applies to a connected component.

Write $\mathcal D_{G,X}(M)$ for the deformation space intended by the source and $\operatorname{Rep}(\pi_1(M),G)$ for its intended representation space. A common marked deformation-space convention identifies structures under pullback by diffeomorphisms isotopic to the identity. Possible target conventions include $\operatorname{Hom}(\pi_1(M),G)$, a conjugacy quotient, and, where applicable, an algebraic character quotient. These are different choices; the supplied record does not identify the intended equivalence relations or topologies. In particular, a map to homomorphisms themselves requires appropriate framing or normalization data.

Source question, with unresolved conventions retained. For every closed manifold $M$ and every geometric model $(G,X)$ within the intended scope, consider the holonomy map \[ \operatorname{Hol}\colon\mathcal D_{G,X}(M)\longrightarrow \operatorname{Rep}(\pi_1(M),G). \] The source describes this map as open with discrete fibers. Treating these as supplied properties in the intended setting, is $\operatorname{Hol}$ always a branched local homeomorphism? Find conditions under which it is a local homeomorphism.

Here openness means that $\operatorname{Hol}(U)$ is open in the target for every open subset $U$ of the domain. Discrete fibers means that each fiber, with its subspace topology, is discrete. Being a local homeomorphism at $s\in\mathcal D_{G,X}(M)$ means that there are open neighborhoods $U$ of $s$ and $V$ of $\operatorname{Hol}(s)$ such that \[ \operatorname{Hol}|_U\colon U\xrightarrow{\ \cong\ }V \] is a homeomorphism. The second request asks for hypotheses ensuring this property at specified structures or throughout the deformation space.

The source does not define "branched local homeomorphism." A precise version of the first question therefore requires a specified branching definition, including any local-model, finiteness, or exceptional-set requirements. No such requirements are inferred here. The source's assertion of openness and discrete fibers also remains qualified by its unspecified deformation-space and representation-space conventions.

Short English statement

For geometric structures on a closed manifold, the source asks whether the holonomy map, described there as open with discrete fibers, always has a branched local homeomorphism structure, and which hypotheses make it a local homeomorphism. The intended spaces, geometric hypotheses, and meaning of branching require clarification before the first question has a unique precise formulation.

Sources

• Local UnsolvedMath registry, permanent ID $30003957$, exact source problem code catalogue entry, title Local Branching of Holonomy Maps for Geometric Structures; collection: Oberwolfach Reports Open Problems. The supplied record labels the problem open and the statement corrected_verified, with a statement-verification note asserting preservation of mathematical content and repair of notation, typography, and omitted definitions. These are supplied registry assessments, not independent verification.
• Supplied report citation: New Trends in Teichmüller Theory and Mapping Class Groups (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/40, pp. 2475--2534. https://doi.org/10.4171/owr/2018/40. The background describes these reports as records of mathematical workshops and their open-problem discussions. The literature-source label is "Oberwolfach Report 40, Alessandrini problem"; no proposer is specified in the proposer field. The supplied proposed year is 2018.
• Supplied literature source: Goldman, geometric structures on manifolds. https://math.umd.edu/~wmg/gstom.pdf.
• Supplied literature source: Projective structures and quasi-Hitchin holonomy. https://doi.org/10.48550/arXiv.2110.15407.
• Supplied literature assessment dated exactly 2026-08-21: the Ehresmann--Thurston principle gives local homeomorphism in standard compact settings, and special loci are understood, while the universal branched-local-homeomorphism claim and general criteria are described as unresolved. This is supplied dated curation derived from the OWR material, not independently verified. The source expressly requests independent verification of time-sensitive status and cited claims.