{"paper":{"title":"The topology of spaces of polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Michael Farber, Viktor Fromm","submitted_at":"2011-05-03T15:48:22Z","abstract_excerpt":"Let $E_{d}(\\ell)$ denote the space of all closed $n$-gons in $\\R^{d}$ (where $d\\ge 2$) with sides of length $\\ell_1,..., \\ell_n$, viewed up to translations. The spaces $E_d(\\ell)$ are parameterized by their length vectors $\\ell=(\\ell_1,..., \\ell_n)\\in \\R^n_{>}$ encoding the length parameters. Generically, $E_{d}(\\ell)$ is a closed smooth manifold of dimension $(n-1)(d-1)-1$ supporting an obvious action of the orthogonal group ${O}(d)$. However, the quotient space $E_{d}(\\ell)/{O}(d)$ (the moduli space of shapes of $n$-gons) has singularities for a generic $\\ell$, assuming that $d>3$; this quot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0613","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}