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Moreover, it is shown if $k$ is the least positive integer that $f$ is represented by a cycle in $\\mathrm{Ext}^{k,k+n}_A(\\Z/2,\\Z/2)$, then (i) if $e_*h(f)\\neq 0$ then $n\\geqslant 2^k-1$; (ii) if $e_*h(f)=0$ then $n\\geqslant 2^k-2^t$ for some $t>1$. Second, for $S\\subseteq{_2\\pi_{*>0}^s}$ we show that: (i) if the conjecture holds on $S$, the"},"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":"1504.06752","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-04-25T19:04:14Z","cross_cats_sorted":[],"title_canon_sha256":"db7b4618840d16a8d49457379fb482f48dbe6a279d2a22e89cb743251038ce43","abstract_canon_sha256":"02ec9d18ddc0b994e040555ff42e35ff024d2b6510f6e630e5d7878ffbb92127"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:51:49.182175Z","signature_b64":"I1Lxgsd8is6GzuQ4Yrc7+w7bRz47XKK0iEkb/F0apwpW4QFkISCk2qx2O8+KDWlaf7l5Gi+f0jdflsm+vqLXCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89aa3296c5cd64aaf140c434e6a81c257c7c8147dc79b038ca0e625686c5d6f4","last_reissued_at":"2026-05-18T01:51:49.181534Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:51:49.181534Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Hurewicz homomorphism on the extensions of ideals in $\\pi_*^s$ and spherical classes in $H_*Q_0S^0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Hadi Zare","submitted_at":"2015-04-25T19:04:14Z","abstract_excerpt":"This is about Curtis conjecture on the image of the Hurewicz map $h:{_2\\pi_*}Q_0S^0\\to H_*(Q_0S^0;\\Z/2)$. 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Second, for $S\\subseteq{_2\\pi_{*>0}^s}$ we show that: (i) if the conjecture holds on $S$, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06752","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":"1504.06752","created_at":"2026-05-18T01:51:49.181617+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.06752v2","created_at":"2026-05-18T01:51:49.181617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06752","created_at":"2026-05-18T01:51:49.181617+00:00"},{"alias_kind":"pith_short_12","alias_value":"RGVDFFWFZVSK","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"RGVDFFWFZVSKV4KA","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"RGVDFFWF","created_at":"2026-05-18T12:29:39.896362+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/RGVDFFWFZVSKV4KAYQ2ONKA4EV","json":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV.json","graph_json":"https://pith.science/api/pith-number/RGVDFFWFZVSKV4KAYQ2ONKA4EV/graph.json","events_json":"https://pith.science/api/pith-number/RGVDFFWFZVSKV4KAYQ2ONKA4EV/events.json","paper":"https://pith.science/paper/RGVDFFWF"},"agent_actions":{"view_html":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV","download_json":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV.json","view_paper":"https://pith.science/paper/RGVDFFWF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.06752&json=true","fetch_graph":"https://pith.science/api/pith-number/RGVDFFWFZVSKV4KAYQ2ONKA4EV/graph.json","fetch_events":"https://pith.science/api/pith-number/RGVDFFWFZVSKV4KAYQ2ONKA4EV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV/action/storage_attestation","attest_author":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV/action/author_attestation","sign_citation":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV/action/citation_signature","submit_replication":"https://pith.science/pith/RGVDFFWFZVSKV4KAYQ2ONKA4EV/action/replication_record"}},"created_at":"2026-05-18T01:51:49.181617+00:00","updated_at":"2026-05-18T01:51:49.181617+00:00"}