Homotopy Between Salvetti and Bessis–Brady–Watt Complexes

Definitions and mathematical statement

Write $S$ for the Salvetti complex and $B$ for the Bessis--Brady--Watt complex intended in the supplied source. Their underlying group or arrangement, precise constructions, and any auxiliary choices are not specified in the supplied excerpt. Consequently, these symbols designate intended objects whose definitions and scope must be recovered before the problem has a fully determined formulation.

Requested construction. Construct an explicit homotopy between $S$ and $B$, in the sense intended by the source.

Unresolved meaning of the request. A homotopy ordinarily relates two maps with the same domain and codomain, rather than two spaces directly. If the intended request is an explicit homotopy equivalence of the two complexes, a precise formulation would be to construct continuous maps \[ f:S\longrightarrow B,\qquad g:B\longrightarrow S \] and continuous homotopies \[ H:S\times[0,1]\longrightarrow S,\qquad K:B\times[0,1]\longrightarrow B \] satisfying, for every $x\in S$ and $y\in B$, \[ H(x,0)=g(f(x)),\quad H(x,1)=x, \qquad K(y,0)=f(g(y)),\quad K(y,1)=y. \] This is a conditional interpretation, not a confirmed replacement of the source's request. The excerpt does not specify maps to be related by a homotopy, a common ambient space, a deformation retraction, or requirements such as cellularity or equivariance. It also does not define what counts as an explicit construction or state the family of groups over which the construction is requested.

Supplied geometric context. The excerpt motivates a triangulation by this homotopy question and asks whether that triangulation has geometric meaning. It contrasts the $n!$ vertices of weak order with the $\operatorname{Cat}(n)$ vertices of a noncrossing partition lattice, then discusses the Tamari lattice and, more generally, Cambrian lattices. The meaning of $n$, the indexing convention for $\operatorname{Cat}(n)$, and the scope of these counts are not supplied; they must not be read as counts for every group denoted by $W$.

The source describes Cambrian lattices as lattice quotients of weak order and reports that, up to taking inverses, elements $w\in W$ in the same Cambrian congruence class have the same concordant chains, so that the same piece of $\mathrm{NC}(W)$ is reused. It further reports that restriction to sortable elements gives a non-convex triangulation of the associahedron. These are contextual claims attributed to the source, not established here. The excerpt does not define concordant chains, the pieces being reused, the relevant Cambrian congruence or sortable elements, or the precise inverse convention. It does not supply a construction connecting these claims to the requested homotopy.

Short English statement

The source asks for an explicit homotopy relating the Salvetti and Bessis--Brady--Watt complexes. An explicit homotopy equivalence is one possible interpretation, but the intended complexes, parameter range, and meaning of the requested homotopy remain unspecified. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006526$, source problem code catalogue entry, title: Homotopy Between Salvetti and Bessis--Brady--Watt Complexes. The supplied record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2026, and names no proposer.
• Supplied report citation: Enumerative Combinatorics (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/2, pp. 81--156. https://doi.org/10.4171/owr/2026/2. These are the supplied report page bounds; no precise problem page or individual speaker attribution is provided.
• Supplied literature source: Williams, Permutahedron Triangulations. https://personal.utdallas.edu/~nxw170830/docs/Bessalvetti/bessalvetti.html. No fuller bibliographic information is supplied.
• Supplied literature assessment dated 2026-08-22: the report explicitly states that an explicit homotopy between the two complexes remains open; later triangulation work provides a framework but not the requested completed homotopy. This assessment and the open label are supplied dated curation, not independently verified here. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that only clear formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. This is supplied editorial metadata and does not resolve the formulation issues above or constitute mathematical verification of this draft.