Homology-Sphere Factors in Simplicial Decompositions

Definitions and mathematical statement

Use the standard finite simplicial-complex convention and let $k$ be a field. For a face $\sigma$ of a complex $K$, its link is \[ \operatorname{lk}_K(\sigma)=\{\tau\in K:\tau\cap\sigma=\varnothing,\ \tau\cup\sigma\in K\}. \] A pure $d$-dimensional complex is Cohen--Macaulay over $k$ when the reduced homology of every link vanishes in degrees below the dimension of that link. It is doubly Cohen--Macaulay when it is Cohen--Macaulay and, for every vertex $v$, the deletion $K-v=\{\sigma\in K:v\notin\sigma\}$ is Cohen--Macaulay of the same dimension $d$.

Let $K$ be such a doubly Cohen--Macaulay complex, with $d>1$. The question asks whether the decomposition supplied by the source's Theorem 7 can be chosen in the form \[ K=\bigcup_{i=1}^m S_i, \] where the $S_i$ are subcomplexes that are homology $d$-spheres and satisfy the specified overlap and reordering properties. In the usual local meaning, a homology $d$-sphere over $k$ is a pure complex $S$ with \[ \widetilde H_j(\operatorname{lk}_S(\sigma);k)\cong \begin{cases}k,&j=d-|\sigma|,\\0,&j\ne d-|\sigma|,\end{cases} \] for every face $\sigma$, including the empty face, using the standard reduced-homology convention in degree $-1$. The source does not specify whether it intends this local condition or only the global homology of a sphere, or whether its sphere coefficients are the same field $k$.

The cleaned reordering condition naturally reads: for every $a\in\{1,\ldots,m\}$, there is a permutation $\pi$ with $\pi(1)=a$ such that for every $j\geq2$, the intersection \[ S_{\pi(j)}\cap\bigcup_{\ell<j}S_{\pi(\ell)} \] contains a $d$-face. This permits the remaining order to depend on the selected first factor; it is not the assertion that every ordering works. However, "intersects … in a $d$-face" could instead require the entire intersection to equal the simplex on one $d$-face, a stronger condition. Theorem 7 is not reproduced, so the draft records this ambiguity rather than declaring these readings equivalent. It also cannot recover any additional minimal-cycle or other conditions imposed by that theorem.

The 2026-08-21 literature assessment reports that choosing all factors as homology spheres remains open, with no later resolution located in its checked literature. The original question is specifically an enhancement of the decomposition in Theorem 7; the cleaned formulation supplies the doubly Cohen--Macaulay and overlap context. Both the dated open-status assessment and reported official-source normalization are retained without independent verification.

Short English statement

Can the factors in the indicated decomposition of every doubly Cohen--Macaulay complex of dimension greater than one all be homology spheres, while preserving its overlap and choice-of-first-factor properties? The missing Theorem 7 leaves important details unresolved, especially whether an overlap contains a facet or equals a single facet.

Sources

• Local UnsolvedMath registry, permanent ID 30000214, source code catalogue entry, "Homology-Sphere Factors in Simplicial Decompositions," including the original reference to Theorem 7 and the expanded cleaned formulation.
• Discrete Geometry (2006), Oberwolfach Reports, pp. 925--994, DOI 10.4171/owr/2005/17: https://doi.org/10.4171/owr/2005/17. The report citation uses 2006, whereas its DOI and proposed-year metadata use 2005.
• Supplied reference "Related minimal-cycle research": https://doi.org/10.1137/23M1607696.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.