Topological Types of Linear-Approach Spaces

Definitions and mathematical statement

Let $X$ be a translation surface, let $d_X$ be its flat length metric, and let $\overline X$ be its metric completion. Fix $x\in\overline X$. A translation structure is given by Euclidean coordinate charts on the regular part of $X$ whose transition maps are translations.

Using an endpoint parametrization, a linear approach to $x$ is represented by a unit-speed geodesic trajectory \[ \gamma:(0,\varepsilon)\longrightarrow X,\qquad \varepsilon>0,\qquad \lim_{t\downarrow0}d_{\overline X}(\gamma(t),x)=0. \] Here the trajectory lies in the regular flat part and is locally a straight line in translation coordinates. Two representatives $\gamma_1:(0,\varepsilon_1)\to X$ and $\gamma_2:(0,\varepsilon_2)\to X$ define the same germ if there exists $0<\delta<\min\{\varepsilon_1,\varepsilon_2\}$ such that \[ \gamma_1(t)=\gamma_2(t)\qquad(0<t<\delta). \] Write $L(x)$ for the set of these germs, equipped with the topology intended in the source report.

Classification question. Characterize, up to homeomorphism, all topological spaces $T$ for which there exist a translation surface $X$ and a point $x\in\overline X$ such that \[ T\cong L(x). \]

Unresolved qualifications. The supplied record does not define the topology on $L(x)$; the germ equivalence relation above specifies only its underlying set. Thus the topological classification target remains conditional on recovering that topology from the cited report. The record also does not specify the permitted class of translation surfaces, including any restrictions on area, genus, compactness, boundary, or completion singularities. These restrictions must be recovered before the range of the existential quantifier over $X$ is fully determined. The endpoint parametrization and regular-trajectory convention above make the usual germ interpretation explicit, but require confirmation against the report.

Short English statement

Determine which topological spaces occur as spaces of germs of geodesic trajectories approaching a point in the metric completion of a translation surface. The topology on the germ space and the precise class of allowed translation surfaces are missing from the supplied record, so a fully specified classification problem requires checking the cited report.

Sources

• Local UnsolvedMath registry, permanent ID $30001787$, title Topological Types of Linear-Approach Spaces, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2011; no individual proposer is identified.
• Billiards, Flat Surfaces, and Dynamics on Moduli Spaces (2011), Oberwolfach Reports, pp. 1361--1427, DOI 10.4171/owr/2011/25. Supplied source report and literature-source link labeled "original question": https://doi.org/10.4171/owr/2011/25.
• The supplied original question is: "Can the topological spaces that arise as L(x) be characterized?" The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This verification is supplied provenance, not an independent verification by this draft.
• Supplied dated literature curation, checked 2026-08-21: status open. The supplied assessment says that the report presents characterization of the possible topological spaces as a future question and that no general classification was found. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.