{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KL2V74NLG2PA7M2CAI6GBCZDMR","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":"1aacf435b8feb7826fc3e541311576349a60750f4999fc05042273385a77feb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-25T21:39:57Z","title_canon_sha256":"8ea0050f8261dff2dc0790a4cdb779c196989c42deacb93ed34569a2078bd44b"},"schema_version":"1.0","source":{"id":"1609.07809","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.07809","created_at":"2026-05-18T00:45:06Z"},{"alias_kind":"arxiv_version","alias_value":"1609.07809v2","created_at":"2026-05-18T00:45:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.07809","created_at":"2026-05-18T00:45:06Z"},{"alias_kind":"pith_short_12","alias_value":"KL2V74NLG2PA","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KL2V74NLG2PA7M2C","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KL2V74NL","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:0a1f3d0915cfa43db91b79a9291564c9f22cdd27b469a4c525075ece2145ad09","target":"graph","created_at":"2026-05-18T00:45:06Z","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":"Given an $L^2$-acyclic connected finite $CW$-complex, we define its universal $L^2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\\operatorname{Wh}^w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincar\\'e duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group $\\operatorname{Wh}^w(G)$ to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal $L^2$-torsion can be identif","authors_text":"Stefan Friedl, Wolfgang L\\\"uck","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-25T21:39:57Z","title":"Universal $L^2$-torsion, polytopes and applications to $3$-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07809","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":"sha256:8e17f36cebb46ab0a7eeddcad138661fd09cbb8b9ad237747f2727ef87e57cc6","target":"record","created_at":"2026-05-18T00:45:06Z","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":"1aacf435b8feb7826fc3e541311576349a60750f4999fc05042273385a77feb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-25T21:39:57Z","title_canon_sha256":"8ea0050f8261dff2dc0790a4cdb779c196989c42deacb93ed34569a2078bd44b"},"schema_version":"1.0","source":{"id":"1609.07809","kind":"arxiv","version":2}},"canonical_sha256":"52f55ff1ab369e0fb342023c608b2364450f1d9206b519764f728282238e58e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"52f55ff1ab369e0fb342023c608b2364450f1d9206b519764f728282238e58e7","first_computed_at":"2026-05-18T00:45:06.376373Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:06.376373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xTa7YaLe6KGpB7tu3ZsWUkyMSAuhh5tax26hYih4VVg8jZXdGcJLVoewkFN4BAlRWBD1DD5ZJyz40HA1MpB1DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:06.376796Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.07809","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8e17f36cebb46ab0a7eeddcad138661fd09cbb8b9ad237747f2727ef87e57cc6","sha256:0a1f3d0915cfa43db91b79a9291564c9f22cdd27b469a4c525075ece2145ad09"],"state_sha256":"ed426535e648fb316d71aacbe5c64e0496c4e9bddd5e46e65f73749c5166a053"}