{
  "annotation_id": "69bd79a2-f3e0-46d9-a1dc-76644ff38cce",
  "date_created": "2026-02-17T05:53:24.412000Z",
  "date_modified": "2026-02-17T05:53:24.412000Z",
  "file_hash": "0851e941e4f9b639bfdb5d9f24f681f44a7dde806a9010176af81cb0f5e92071",
  "private": false,
  "record": {
    "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 $g_3=1$.",
    "arxiv_id": "2601.10072",
    "authors": [
      "Isabella Novik",
      "Hailun Zheng"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Simplicial spheres with $g_k=1$",
    "url": "https://arxiv.org/abs/2601.10072",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "6d4b70fe-32cd-43de-9c5b-4689041b1aec",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}