Simplicial spheres with $g_k=1$

TL;DR

This paper characterizes simplicial spheres with a specific combinatorial invariant $g_k=1$, extending previous classifications for cases with $g_k=0$ and addressing new conditions for missing faces.

Contribution

It provides a comprehensive classification of simplicial spheres with $g_k=1$ under various dimension and missing face constraints, generalizing earlier results.

Findings

Characterized all simplicial spheres with $g_k=1$ for $d eq 2k$.

Extended classification to the case $d=2k$ with an additional missing face condition.

Achieved a full characterization for 5-spheres with $g_3=1$ without extra assumptions.

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 $g_3=1$.