{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OOR7FVS3RINS5JX3YI6SH3IO4R","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":"9c91f8e4733c8fe31d9f0100168ebe010a8549751c237d890c40460afa4ef566","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-07-28T15:50:29Z","title_canon_sha256":"51420b4f1b335510526a67816cd383ea321f296eae2c1314e4eb3336f220d4c5"},"schema_version":"1.0","source":{"id":"1407.7454","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.7454","created_at":"2026-05-18T00:50:10Z"},{"alias_kind":"arxiv_version","alias_value":"1407.7454v3","created_at":"2026-05-18T00:50:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.7454","created_at":"2026-05-18T00:50:10Z"},{"alias_kind":"pith_short_12","alias_value":"OOR7FVS3RINS","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OOR7FVS3RINS5JX3","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OOR7FVS3","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:ae2d00975b6ee7bb080886619afc90b866f2f921c1435cf6c8cb458bbc607f34","target":"graph","created_at":"2026-05-18T00:50:10Z","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":"We compute the eta function $\\eta(s)$ and its corresponding $\\eta$-invariant for the Atiyah-Patodi-Singer operator $\\mathcal{D}$ acting on an orientable compact flat manifold of dimension $n =4h-1$, $h\\ge 1$, and holonomy group $F\\simeq \\mathbb{Z}_{2^r}$, $r\\in \\mathbb{N}$. We show that $\\eta(s)$ is a simple entire function times $L(s,\\chi_4)$, the $L$-function associated to the primitive Dirichlet character modulo 4. The $\\eta$-invariant is 0 or equals $\\pm 2^k$ for some $k\\ge 0$ depending on $r$ and $n$. Furthermore, we construct an infinite family $\\mathcal{F}$ of orientable $\\mathbb{Z}_{2^","authors_text":"Ricardo A. Podest\\'a","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-07-28T15:50:29Z","title":"The eta function and eta invariant of $\\mathbb{Z}_{2^r}$-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7454","kind":"arxiv","version":3},"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":"sha256:0060dfc14dcbf2f8710136230efa94779e2973ca14ca10a341422f96fd14f7c9","target":"record","created_at":"2026-05-18T00:50:10Z","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":"9c91f8e4733c8fe31d9f0100168ebe010a8549751c237d890c40460afa4ef566","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-07-28T15:50:29Z","title_canon_sha256":"51420b4f1b335510526a67816cd383ea321f296eae2c1314e4eb3336f220d4c5"},"schema_version":"1.0","source":{"id":"1407.7454","kind":"arxiv","version":3}},"canonical_sha256":"73a3f2d65b8a1b2ea6fbc23d23ed0ee4538e747ec29ee4e3b198c403464fa45a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73a3f2d65b8a1b2ea6fbc23d23ed0ee4538e747ec29ee4e3b198c403464fa45a","first_computed_at":"2026-05-18T00:50:10.030482Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:10.030482Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YoIH5OcQUcFZ6a+ji561vuAFkHFUWyI3ZXXlC/t5XFoDyPBBddJiurRrBSC9v5epJBZLeBWZ6UA4yFQd45VLCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:10.030993Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.7454","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0060dfc14dcbf2f8710136230efa94779e2973ca14ca10a341422f96fd14f7c9","sha256:ae2d00975b6ee7bb080886619afc90b866f2f921c1435cf6c8cb458bbc607f34"],"state_sha256":"8a761c3dcdd27e544d80507576afdf2159b16ef17978f644454cf8518ba1c109"}