Ball-Product Structure of Cross-Polytope Subcomplexes

Definitions and mathematical statement

For a positive integer $d$, let \[ C_d=\operatorname{conv}\{e_1,-e_1,\ldots,e_d,-e_d\}\subset{\mathbb{R}}^d \] be the $d$-dimensional cross-polytope, where $e_1,\ldots,e_d$ are the standard basis vectors. Its boundary complex has one facet \[ F_w=\{v_1(w),\ldots,v_d(w)\} \] for each word $w=(w_1,\ldots,w_d)\in\{x,y\}^d$, with $v_j(w)=e_j$ if $w_j=x$ and $v_j(w)=-e_j$ if $w_j=y$. Under the usual linear-word convention, define the number of switches by \[ s(w)=\left|\{j\in\{1,\ldots,d-1\}:w_j\ne w_{j+1}\}\right|. \] In particular, this convention does not compare $w_d$ with $w_1$. Define the generated simplicial subcomplex by \[ B(i,d)=\{\sigma\subseteq F_w:\ w\in\{x,y\}^d,\ s(w)\le i\}. \] The supplied record does not explicitly specify the encoding convention or the meaning of a switch; the conventions above make the construction explicit and require confirmation against the source.

For $n\ge0$, let $S^n=\{z\in{\mathbb{R}}^{n+1}:\|z\|=1\}$ and let $B^n=\{z\in{\mathbb{R}}^n:\|z\|\le1\}$ be the closed ball, with $B^0$ a point. The source question is whether $B(i,d)$ is a combinatorial triangulation of \[ S^i\times B^{d-i-1}. \] Interpreting "combinatorial triangulation" in the piecewise-linear sense, this asks whether there exists a PL homeomorphism \[ |B(i,d)|\;\cong_{\mathrm{PL}}\;S^i\times B^{d-i-1}, \] where the product carries its standard PL structure. This interpretation also requires source confirmation.

Unresolved parameter qualification. The supplied statement gives neither a parameter range nor an explicit quantifier over $i$ and $d$. The displayed sphere and ball have their ordinary meanings for integers $d\ge1$ and $0\le i\le d-1$, but these minimal dimensional restrictions do not establish the intended range of the source question. In particular, whether the question quantifies over every such pair, a restricted subrange, or specified endpoint cases remains unresolved.

Short English statement

The question asks whether the cross-polytope boundary subcomplex generated by facets with at most $i$ switches triangulates the product of an $i$-sphere and a closed $(d-i-1)$-ball in the combinatorial, interpreted here as PL, sense. The intended parameter range, encoding convention, and precise triangulation terminology require source confirmation.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001696$, source problem code catalogue entry, title "Ball-Product Structure of Cross-Polytope Subcomplexes." The supplied record assigns the problem to the Oberwolfach Reports Open Problems collection and gives proposed year 2011; no individual proposer is supplied. Its status is recorded as open, and its statement status as corrected_verified. These are supplied registry labels, not independent verification by this draft.
• Supplied report citation: Topological and Geometric Combinatorics (2011), Oberwolfach Reports, pp. 349--423, DOI 10.4171/owr/2011/08. Source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2011/08. No precise problem page or verbatim report passage is supplied.
• Supplied literature-source link labelled "Klee--Novik centrally symmetric manifolds": https://arxiv.org/abs/1102.0542. The supplied assessment attributes to Klee--Novik a proof that the boundary of $B(i,d)$ triangulates a product of spheres, while describing the stronger ball-product triangulation question as open in the located literature. The record does not give the hypotheses or precise theorem statement for that boundary result.
• Literature assessment checked at 2026-08-21: this exact date records supplied dated curation, not an independently verified literature assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims. No source was visited or independently checked in preparing this statement-only draft.