{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:SXAQNM72E4KBV2FCFU5PPFUEJT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"871cfd926c727f0bbfa82b7219720294482d35dfb9c201b658d3e4120c5992a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-12-20T18:41:58Z","title_canon_sha256":"80449e22e7fa7e5944cae1c22144c86fa67456856e69ff20d0af6112325b4044"},"schema_version":"1.0","source":{"id":"1712.07633","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.07633","created_at":"2026-05-18T00:27:33Z"},{"alias_kind":"arxiv_version","alias_value":"1712.07633v1","created_at":"2026-05-18T00:27:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.07633","created_at":"2026-05-18T00:27:33Z"},{"alias_kind":"pith_short_12","alias_value":"SXAQNM72E4KB","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SXAQNM72E4KBV2FC","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SXAQNM72","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:e772256adf1a200654da05fda649da81e7510ada6e0ca81fa54d027c4f4c839e","target":"graph","created_at":"2026-05-18T00:27:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this note we show that a $p$-complete nilpotent space $X$ has a $p$-complete suspension spectrum if and only if its homotopy groups $\\pi_*X$ are bounded $p$-torsion. In contrast, if $\\pi_*X$ is not all bounded $p$-torsion, we locate uncountable rational vector spaces in the integral homology and in the stable homotopy groups of $X$. To prove this, we establish a homological criterion for $p$-completeness of connective spectra. Moreover, we illustrate our results by studying the stable homotopy groups of $K(\\mathbb{Z}_p,n)$ via Goodwillie calculus.","authors_text":"A. K. Bousfield, Tobias Barthel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-12-20T18:41:58Z","title":"On the comparison of stable and unstable $p$-completion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.07633","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b4ae6b36d5c7c10d193cd46e014fd072fa4716be4aac3f71125b2bb89574079c","target":"record","created_at":"2026-05-18T00:27:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"871cfd926c727f0bbfa82b7219720294482d35dfb9c201b658d3e4120c5992a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-12-20T18:41:58Z","title_canon_sha256":"80449e22e7fa7e5944cae1c22144c86fa67456856e69ff20d0af6112325b4044"},"schema_version":"1.0","source":{"id":"1712.07633","kind":"arxiv","version":1}},"canonical_sha256":"95c106b3fa27141ae8a22d3af796844cf5da40b55eb408ede112a7c39676c809","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95c106b3fa27141ae8a22d3af796844cf5da40b55eb408ede112a7c39676c809","first_computed_at":"2026-05-18T00:27:33.984883Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:33.984883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W4486cudKTruVw4FYQ84qgrLbsgSm9NIZTMctkA0FKLMc1DNRsb0nSP1NY22VBCYWSShz8XTGGcCkPLeHq/hBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:33.985827Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.07633","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4ae6b36d5c7c10d193cd46e014fd072fa4716be4aac3f71125b2bb89574079c","sha256:e772256adf1a200654da05fda649da81e7510ada6e0ca81fa54d027c4f4c839e"],"state_sha256":"b5aa5505c0b02e38de1c0509ccbaf2f4a22651cd18b8447ae3b9ffdd189acaca"}