{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:OSBGTNJTNS7CFH5P4ZLCDGTSJN","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":"1c37c83435c12053a0b15c0a4b1a03bccf459344b5a20d0cb65f49d8e04781b2","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-04-02T20:14:53Z","title_canon_sha256":"19c65004e62cfadb3220d2ac7bb75de3c37bc9cb64ca6b2699e0f132fcc3871e"},"schema_version":"1.0","source":{"id":"0804.0427","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.0427","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"arxiv_version","alias_value":"0804.0427v2","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.0427","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"pith_short_12","alias_value":"OSBGTNJTNS7C","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"OSBGTNJTNS7CFH5P","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"OSBGTNJT","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:b8ce181ccdfc9015917d994fdb86480a4613cc9023d4001157b1887ab3e8dcc5","target":"graph","created_at":"2026-05-18T01:22:31Z","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 paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurability class of normal subgroups of G. In particular, we prove that E^n/G is a fiber bundle, with totally geodesic fibers, over a b-dimensional torus, where b is the first Betti number of G.\n  Let N be a normal subgroup of G which is maximal in its commensurability class. We study th","authors_text":"John G. Ratcliffe, Steven T. Tschantz","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-04-02T20:14:53Z","title":"Fibered orbifolds and crystallographic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.0427","kind":"arxiv","version":2},"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":"sha256:68498465851622e4febd3d92f3b29757f572d45ef70fdaa5f144bec381a06414","target":"record","created_at":"2026-05-18T01:22:31Z","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":"1c37c83435c12053a0b15c0a4b1a03bccf459344b5a20d0cb65f49d8e04781b2","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-04-02T20:14:53Z","title_canon_sha256":"19c65004e62cfadb3220d2ac7bb75de3c37bc9cb64ca6b2699e0f132fcc3871e"},"schema_version":"1.0","source":{"id":"0804.0427","kind":"arxiv","version":2}},"canonical_sha256":"748269b5336cbe229fafe656219a724b705da7d122443053c89538929737e86a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"748269b5336cbe229fafe656219a724b705da7d122443053c89538929737e86a","first_computed_at":"2026-05-18T01:22:31.105176Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:31.105176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IdbPRs6VgASIKTe15PobsdEgTaAYf8E4+2pfKZoOi5ScaAX7JyrkMz9SHn8MkbEK/n7hQkdHN+WDuAp7ZNpsCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:31.105709Z","signed_message":"canonical_sha256_bytes"},"source_id":"0804.0427","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:68498465851622e4febd3d92f3b29757f572d45ef70fdaa5f144bec381a06414","sha256:b8ce181ccdfc9015917d994fdb86480a4613cc9023d4001157b1887ab3e8dcc5"],"state_sha256":"fd0a6039c64c2f08f9d00f30c19a0e32264aedd4548b484a69190883407e2df1"}