{"paper":{"title":"Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.AT","math.CO"],"primary_cat":"math.GT","authors_text":"A. Skopenkov","submitted_at":"2021-12-06T20:11:30Z","abstract_excerpt":"Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\\ge3$ odd $K$ embeds into $M$ if and only if there are\n  $\\bullet$ a skew-symmetric $n\\times n$-matrix $A$ with integer entries, whose rank over $\\mathbb Q$ does not exceed $rk H_k(M;\\mathbb Z)$,\n  $\\bullet$ a general position PL map $f:K\\to\\mathbb R^{2k}$, and\n  $\\bullet$ orientations on $k$-faces of $K$ such that for any nonadjacent $k$-faces $\\sigma,\\tau$ of $K$ the entry $A_{\\sigma,\\tau}$ equals to the algebraic intersection"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.06636","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.06636/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}