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Indecomposability in all but these dimensions was obtained by showing that a nontrivial decomposition of $\\Omega S^{2n+1}\\{p\\}$ implies the existence of a $p$-primary Kervaire invariant one element of order $p$ in $\\pi_{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for $\\Omega S^{2n+1}\\{p\\}$ is equivalent"},"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":"1908.05302","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-08-14T18:41:52Z","cross_cats_sorted":[],"title_canon_sha256":"0511e8874fc262b552583034fa0188d7246719cd9e0a5cc604a345a7f88e2d5a","abstract_canon_sha256":"ed3bf5513d90769e64d933511c179fa36b97f6a6d8514c59f2c6e9366e8dd813"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:29:38.647491Z","signature_b64":"iXb3bPL0E3zCz0SLMGey6Rgl2Eoiak2Ttd4IGebWxjG3ibx1V0aM6l13efGzjH4CCpO+xzW8y1HZx6DkOLGdCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbc35f0f0e1d952f9f3da15e13c9c78cbb4c9f07d075afffff450bcf6e79dca3","last_reissued_at":"2026-07-05T03:29:38.646929Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:29:38.646929Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The fibre of the degree $3$ map, Anick spaces and the double suspension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Steven Amelotte","submitted_at":"2019-08-14T18:41:52Z","abstract_excerpt":"Let $S^{2n+1}\\{p\\}$ denote the homotopy fibre of the degree $p$ self map of $S^{2n+1}$. 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