Simplicial spheres with $g_k=1$

TL;DR

The paper studies simplicial $(d-1)$-spheres with no large missing faces that satisfy $g_k=1$ for $k\ge 2$. Building on the generalized lower bound framework and stress-space techniques, the authors provide a complete characterization for $d\ge 2k+1$ and the additional $d=2k$ scenario, showing that such spheres are precisely joins of boundary spheres of simplices as described in the main theorem. The analysis hinges on an enhanced cone-lemma for stresses and McMullen’s integral formula to relate local link data to global structure, yielding exact decompositions into joins like $\partial\sigma^{d-k} * \Gamma$ or $\partial\sigma^j * \partial\sigma^{d-j}$. For the special case $k=3$, $d=5$, they completely classify all $5$-spheres in $S(2,5)$ with $g_3=1$, proving these are boundaries of simplicial polytopes and are joins of three 3-cycles. The work advances understanding of the GLBC equality cases in small $g_k$ regimes and connects combinatorial geometry with stress-space methods, offering both structural results and open problems for broader missing-face configurations.

Abstract

For $d\geq 4$, Kalai (1987) characterized all simplicial $(d-1)$-spheres with $g_2=0$, and for $k\geq 2$ and $d\geq 2k$, Murai and Nevo (2013) characterized all simplicial $(d-1)$-spheres with $g_k=0$. In addition, for $d\geq 4$, Nevo and Novinsky (2011) characterized all simplicial $(d-1)$-spheres with $g_2=1$. Motivated by these results, we characterize, for any $k\geq 2$ and $d\geq 2k+1$, all simplicial $(d-1)$-spheres with no missing faces of dimension larger than $d-k$ that satisfy $g_k=1$. When $d=2k$, we obtain a characterization of simplicial $(d-1)$-spheres with $g_k=1$ and no missing faces of dimension greater than $k$, under the additional assumption that there exists at least one missing face of dimension $k$. Finally, for $k=3$, we are able to remove this assumption and characterize all simplicial $5$-spheres with no missing faces of dimension larger than $3$ that satisfy $g_3=1$.

Theorems & Definitions (40)

• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Proposition 2.5
• Proposition 2.6
• Remark 2.7
• Corollary 2.8
• Theorem 2.9
• Lemma 3.1