Research Problems in Function Theory — Problem 7.66

Definitions and mathematical statement

Let $B^n=\{x\in{\mathbb{R}}^n:|x|<1\}$ and let $f:B^n\to{\mathbb{R}}^n$ be continuous. Its branch set in the source's topological sense is \[ B_f=\{x\in B^n:f\text{ is not a local homeomorphism at }x\}. \] A local homeomorphism means that some neighborhood of the point is mapped homeomorphically onto a neighborhood in the target. Compactness of $B_f$ means compactness as a subset of $B^n$, not just boundedness in the ambient Euclidean space.

The printed definition of properness says that $f^{-1}(K)$ is compact in $B^n$ whenever $K$ is a proper subset of $f(B^n)$. This is not the standard compact-preimage condition and quantifies over noncompact and nonclosed subsets as well. A natural possible correction is to require compact preimages for every compact subset $K\subset f(B^n)$, which defines properness relative to the image; properness into all of ${\mathbb{R}}^n$ is another target convention. No replacement is silently made.

Vuorinen's proposed implication, with the source terms retained, is \[ \forall n\ge3\quad \left[f:B^n\to{\mathbb{R}}^n\text{ is continuous, proper, and quasi-analytic, and }B_f\text{ is compact}\right] \Longrightarrow f\text{ is injective}. \] The extract does not define "quasi-analytic", so the admissible class cannot be recovered precisely. In particular it is not silently replaced by quasiregular, quasiconformal, real-analytic, or a particular smooth quasianalytic class. The implication is therefore a literal conditional schema pending the class and properness definitions.

The source says the planar map $z\mapsto z^2$ on $B^2=\mathbb D$ rules out the conclusion in dimension two, and reports the conjecture true when $f(B^n)=B^n$ and $n\ge3$. These attributions require the intended, corrected definitions; the planar example does not validate the literal nonstandard proper-subset condition. The more general open problem mentioned in the final sentence has a missing reference. The supplied triage labels the question open as of 2018 and reports no definitive later resolution, without independent verification here.

Short English statement

Must an appropriate proper map of a ball in dimension at least three be injective when its branch set is compact? The source's properness definition says proper subset where compact subset may have been intended, and its term quasi-analytic is undefined. The two-dimensional counterexample and special onto-ball result therefore remain attributed context until the intended hypotheses are recovered.

Sources

• Local source registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $2307066$, source code catalogue entry, including the supplied OPEN-TRIAGE assessment. Dataset attribution: Ulam AI and UnsolvedMath Contributors; original-source attribution and rights are retained.
• Hayman, Research Problems in Function Theory, source-list date 2018, Problem 7.66, attributed to M. Vuorinen: https://arxiv.org/abs/1809.07200. The metadata reports source-TeX extraction and July 29, 2026 access, with current-status and rights review flags. The extracted hypothesis-list markers are corrupted, but all listed conditions are retained.
• The planar example, onto-ball special case, and reference to a more general open problem are supplied without complete separate citations; the last citation appears only as "in ." No linked source was inspected for this draft.