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We work at the prime $2$, with $k=0$, and determine the image for various in the following cases : (1) $m=n$ and $l>0$ arbitrary; (2) $m>n$ and $l=1$. We observe that in most of the cases the image is trivial with the exceptions corresponding to the cases when either there is a (commutative) $H$-space structure on $S^n$ or there is a Hopf invariant one element."},"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":"1806.07079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-06-19T07:32:06Z","cross_cats_sorted":[],"title_canon_sha256":"3b651c456a0bea2ec6e6fdfee95d70d839764dd03cf06d3154d2c88c5b4e6648","abstract_canon_sha256":"5ae2f4b3b974243ceaa864844ab839e98f858704ca96a93c31033e17feee6ae3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:28.358727Z","signature_b64":"AeA2Ug8l5TVi8F47cBjJcyvmlJu0we0FzlMUvUVmz/AZ6slclyqVqnL8HXfXOOXYkjPmp0XP9QxJjpD/e7NzCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e33cd5d4b151eea4449037a814f5be33d4f7b5581dadf670f99c65ac609a05c5","last_reissued_at":"2026-05-17T23:43:28.358216Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:28.358216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the image of the unstable Boardman map","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Hadi Zare","submitted_at":"2018-06-19T07:32:06Z","abstract_excerpt":"We consider the `unstable Boardman map' (homomorphism if $k>0$) $$b:\\pi^{m+k}\\Sigma^k\\Omega^lS^{n+l}\\simeq[\\Omega^lS^{n+l},\\Omega^kS^{m+k}]\\longrightarrow \\mathrm{Hom}(H_*\\Omega^lS^{n+l},H_*\\Omega^kS^{m+k})$$ defined by $h(f)=f_*$. We work at the prime $2$, with $k=0$, and determine the image for various in the following cases : (1) $m=n$ and $l>0$ arbitrary; (2) $m>n$ and $l=1$. We observe that in most of the cases the image is trivial with the exceptions corresponding to the cases when either there is a (commutative) $H$-space structure on $S^n$ or there is a Hopf invariant one element."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07079","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1806.07079","created_at":"2026-05-17T23:43:28.358287+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.07079v2","created_at":"2026-05-17T23:43:28.358287+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.07079","created_at":"2026-05-17T23:43:28.358287+00:00"},{"alias_kind":"pith_short_12","alias_value":"4M6NLVFRKHXK","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"4M6NLVFRKHXKIREQ","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"4M6NLVFR","created_at":"2026-05-18T12:32:05.422762+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP","json":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP.json","graph_json":"https://pith.science/api/pith-number/4M6NLVFRKHXKIREQG6UBJ5N6GP/graph.json","events_json":"https://pith.science/api/pith-number/4M6NLVFRKHXKIREQG6UBJ5N6GP/events.json","paper":"https://pith.science/paper/4M6NLVFR"},"agent_actions":{"view_html":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP","download_json":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP.json","view_paper":"https://pith.science/paper/4M6NLVFR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.07079&json=true","fetch_graph":"https://pith.science/api/pith-number/4M6NLVFRKHXKIREQG6UBJ5N6GP/graph.json","fetch_events":"https://pith.science/api/pith-number/4M6NLVFRKHXKIREQG6UBJ5N6GP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP/action/storage_attestation","attest_author":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP/action/author_attestation","sign_citation":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP/action/citation_signature","submit_replication":"https://pith.science/pith/4M6NLVFRKHXKIREQG6UBJ5N6GP/action/replication_record"}},"created_at":"2026-05-17T23:43:28.358287+00:00","updated_at":"2026-05-17T23:43:28.358287+00:00"}