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As an application we provide a new simplified proof of the fact that $\\pi_n^{Lip}(H_n)\\neq 0$, $n=1,2,...$, and we prove a new result that $\\pi_{4n-1}^{Lip}(H_{2n})\\neq 0$ for $n=1,2,...$ The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space $W^{1,p}(M,H_{2n})$ when $dim M\\geq 4n$ and $4n-1\\"},"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":"1301.4978","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-01-21T20:45:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"c749adb01fe329f14407682833de1081027b1813f76ef475914a0340faf449da","abstract_canon_sha256":"24fff5d364e7324d69a846c6a1b2901fcaf7654c4c4ba816b301ff9a8420b3e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:03.477146Z","signature_b64":"qyeRzXjn4/LSELFa/1JrifvHZVe4pJKmdquSPcMiFSDTUolRLSJmbv/6Jh3lQWKnx0iLTVkt5fj+pHXvdrdoBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3abf94e5ae0e5f7d4b723cb551aa34f2719406538323fb4ae491a9af1530946e","last_reissued_at":"2026-05-18T03:06:03.476689Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:03.476689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GT","authors_text":"Armin Schikorra, Jeremy T. 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