Questions in Geometric Group Theory — Q 13.7

Definitions and mathematical statement

Let $k\geq1$ and let $L$ be a finite simplicial complex whose underlying topological space is homeomorphic to the sphere $S^{2k-1}$. The complex is flag if every finite set of vertices which are pairwise joined by edges spans a simplex. This condition rules out a collection of mutually adjacent vertices whose expected simplex is missing.

For $0\leq i\leq2k-1$, let $f_i(L)$ be the number of $i$-dimensional simplices, counted without orientations. Define the rational number \[ C(L)=1-\sum_{i=0}^{2k-1}(-1)^i\frac{f_i(L)}{2^{i+1}}. \] The Charney--Davis question asks whether every such flag triangulation satisfies \[ (-1)^k C(L)\geq0. \] The initial constant $1$ is part of this expression; $C(L)$ is not simply the ordinary Euler characteristic of the sphere. The exponent $k$ is half of one more than the sphere dimension, exactly as displayed.

Both the integer $k$ and the finite triangulation vary universally. No assumption that the sphere is the boundary of a convex polytope is made, and no particular flag subdivision is prescribed. A proof for polytopal spheres or low dimensions would therefore cover only a restricted family. The assertion is non-strict, so equality is permitted. All the quantities depend on the simplicial face numbers, and the problem asks for a numerical inequality valid for every permissible combinatorial realization of the sphere, rather than invariance of this number under arbitrary subdivisions.

Short English statement

Does every flag triangulation of an odd-dimensional sphere satisfy the signed Charney–Davis face-number inequality?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101307 (AMR-010-1307), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 13.7 (PDF page 25). Original source indexed by AMR-010-1307. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.