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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1105.0613","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-05-03T15:48:22Z","cross_cats_sorted":[],"title_canon_sha256":"082f3a1698472ae99a34bdef6db3f8a3a10b6ec1a97a1c8ac671d6082cc4f031","abstract_canon_sha256":"76e5ccde68c514bcef3c0ba721cc576b0f81528827d1de159e97bd977e5960d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:11.013407Z","signature_b64":"EDpxXKEDsjMZ+PihoHEaGngOvwPygA6xCHNTWKyooBv8SIoiLH0xmqIzNx6Q8XXAgQim3HrOzAvmf+kxodMeDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07affe1cc4524952dd9a3b19f6d49213c38c0cf38659276f93526dcfdb360012","last_reissued_at":"2026-05-18T04:23:11.012981Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:11.012981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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