{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:NUQ2ABQMY5PVFGAZJADDUKQUXG","short_pith_number":"pith:NUQ2ABQM","canonical_record":{"source":{"id":"1605.02387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-05-08T23:27:42Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6fbe3749404b6264441abf70df54683cea48d60e9c2854b865171093009ff3b7","abstract_canon_sha256":"2fbd7a2ce9d83960d70af25d4251be5c1638142d5e32930fe81c1bfd42b9a5f9"},"schema_version":"1.0"},"canonical_sha256":"6d21a0060cc75f52981948063a2a14b988b0a9ae7140a4ab879a7aa86754f3e0","source":{"kind":"arxiv","id":"1605.02387","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.02387","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"arxiv_version","alias_value":"1605.02387v1","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.02387","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"pith_short_12","alias_value":"NUQ2ABQMY5PV","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"NUQ2ABQMY5PVFGAZ","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"NUQ2ABQM","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:NUQ2ABQMY5PVFGAZJADDUKQUXG","target":"record","payload":{"canonical_record":{"source":{"id":"1605.02387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-05-08T23:27:42Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6fbe3749404b6264441abf70df54683cea48d60e9c2854b865171093009ff3b7","abstract_canon_sha256":"2fbd7a2ce9d83960d70af25d4251be5c1638142d5e32930fe81c1bfd42b9a5f9"},"schema_version":"1.0"},"canonical_sha256":"6d21a0060cc75f52981948063a2a14b988b0a9ae7140a4ab879a7aa86754f3e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:21.131328Z","signature_b64":"ffDsXhy91XBf7AwItXvM42nqTQrrJZQhwVPIeGRrie7S5FjkwZDt/mg2xYbXNfwS0QSSN+KCoWj9CoUuHlW6Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d21a0060cc75f52981948063a2a14b988b0a9ae7140a4ab879a7aa86754f3e0","last_reissued_at":"2026-05-18T01:15:21.130579Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:21.130579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.02387","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:15:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KwND9ejPDi7hYN3VTFkuiEt470wrSczdULFRoTGJk5PaPoXvyTgKi75/3B7Csi3r78MUMqBK83qoIz6lP5lUCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:38:07.855346Z"},"content_sha256":"2aa501b3537f974b20c5a279a36dab179581c1a331399b859e1af32370efd0ee","schema_version":"1.0","event_id":"sha256:2aa501b3537f974b20c5a279a36dab179581c1a331399b859e1af32370efd0ee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:NUQ2ABQMY5PVFGAZJADDUKQUXG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spheroidal groups, virtual cohomology and lower dimensional G-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AT","authors_text":"William Browder","submitted_at":"2016-05-08T23:27:42Z","abstract_excerpt":"A space is defined to be \"$n$-spheroidal\" if it has the homotopy type of an $n$-dimensional CW-complex $X$ with $H_{n}(X, \\mathbb{Z})$ not zero and finitely generated. A group $G$ is called \"$n$-spheroidal\" if its classifying space $K(G,1)$ is $n$-spheroidal. Examples include fundamental groups of compact manifold $K(G,1)$'s. Moreover, the class of groups $G$ which are $n$-spheroidal for some $n$, is closed under products, free products, and group extensions. If $Y$ is a space with $\\pi_{1}(Y)$ $n$-spheroidal, and if $H_{k}(Y;\\mathbb{F}_{p})$ is non-zero and finitely generated, and if $H_{i}(Y"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02387","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:15:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EPqpiitWcss/e3Dgp43/pkosd/dcjxXaJ/aX/ycHy0UTqIG4xfMGN2on8GUH3NaOXORSPxVVu8aRY71b0dozDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:38:07.855699Z"},"content_sha256":"53fa891bbe90123e7385b67127ab318609aa3da08933a6e022c30f6c90027177","schema_version":"1.0","event_id":"sha256:53fa891bbe90123e7385b67127ab318609aa3da08933a6e022c30f6c90027177"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NUQ2ABQMY5PVFGAZJADDUKQUXG/bundle.json","state_url":"https://pith.science/pith/NUQ2ABQMY5PVFGAZJADDUKQUXG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NUQ2ABQMY5PVFGAZJADDUKQUXG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T19:38:07Z","links":{"resolver":"https://pith.science/pith/NUQ2ABQMY5PVFGAZJADDUKQUXG","bundle":"https://pith.science/pith/NUQ2ABQMY5PVFGAZJADDUKQUXG/bundle.json","state":"https://pith.science/pith/NUQ2ABQMY5PVFGAZJADDUKQUXG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NUQ2ABQMY5PVFGAZJADDUKQUXG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:NUQ2ABQMY5PVFGAZJADDUKQUXG","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":"2fbd7a2ce9d83960d70af25d4251be5c1638142d5e32930fe81c1bfd42b9a5f9","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-05-08T23:27:42Z","title_canon_sha256":"6fbe3749404b6264441abf70df54683cea48d60e9c2854b865171093009ff3b7"},"schema_version":"1.0","source":{"id":"1605.02387","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.02387","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"arxiv_version","alias_value":"1605.02387v1","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.02387","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"pith_short_12","alias_value":"NUQ2ABQMY5PV","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"NUQ2ABQMY5PVFGAZ","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"NUQ2ABQM","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:53fa891bbe90123e7385b67127ab318609aa3da08933a6e022c30f6c90027177","target":"graph","created_at":"2026-05-18T01:15:21Z","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":"A space is defined to be \"$n$-spheroidal\" if it has the homotopy type of an $n$-dimensional CW-complex $X$ with $H_{n}(X, \\mathbb{Z})$ not zero and finitely generated. A group $G$ is called \"$n$-spheroidal\" if its classifying space $K(G,1)$ is $n$-spheroidal. Examples include fundamental groups of compact manifold $K(G,1)$'s. Moreover, the class of groups $G$ which are $n$-spheroidal for some $n$, is closed under products, free products, and group extensions. If $Y$ is a space with $\\pi_{1}(Y)$ $n$-spheroidal, and if $H_{k}(Y;\\mathbb{F}_{p})$ is non-zero and finitely generated, and if $H_{i}(Y","authors_text":"William Browder","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-05-08T23:27:42Z","title":"Spheroidal groups, virtual cohomology and lower dimensional G-spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02387","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":"sha256:2aa501b3537f974b20c5a279a36dab179581c1a331399b859e1af32370efd0ee","target":"record","created_at":"2026-05-18T01:15:21Z","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":"2fbd7a2ce9d83960d70af25d4251be5c1638142d5e32930fe81c1bfd42b9a5f9","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-05-08T23:27:42Z","title_canon_sha256":"6fbe3749404b6264441abf70df54683cea48d60e9c2854b865171093009ff3b7"},"schema_version":"1.0","source":{"id":"1605.02387","kind":"arxiv","version":1}},"canonical_sha256":"6d21a0060cc75f52981948063a2a14b988b0a9ae7140a4ab879a7aa86754f3e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d21a0060cc75f52981948063a2a14b988b0a9ae7140a4ab879a7aa86754f3e0","first_computed_at":"2026-05-18T01:15:21.130579Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:21.130579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ffDsXhy91XBf7AwItXvM42nqTQrrJZQhwVPIeGRrie7S5FjkwZDt/mg2xYbXNfwS0QSSN+KCoWj9CoUuHlW6Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:21.131328Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.02387","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2aa501b3537f974b20c5a279a36dab179581c1a331399b859e1af32370efd0ee","sha256:53fa891bbe90123e7385b67127ab318609aa3da08933a6e022c30f6c90027177"],"state_sha256":"98973b58cb51d2a2cca5a894f6d089da63c7e6f4bcdc54f93ca973e9fd07da40"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tMhxbW85F8cAJggI5M/CtlywnaW6DPtAOLZjChSnNqgmGSppab89C+YUxV6Adc6SWhASF/+1aDrFbTD/mzUqCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T19:38:07.857573Z","bundle_sha256":"d2d216e4332bc5bc5aae8b5578e167725b279beb597571cdd4275f3f527d7248"}}