{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:OPCR27HNDGRRBEXQAFAO3CTFHZ","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":"6a4fe682430b9e1ddd456fe7f5e03120c563a9828c7dd5cc626c1a67f0f503c2","cross_cats_sorted":["math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-06-17T16:17:00Z","title_canon_sha256":"8e6ea94b1de9ff63e44b2d4246e582c19a472d1e029b02ab11085af36ae1836d"},"schema_version":"1.0","source":{"id":"1306.3915","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.3915","created_at":"2026-05-18T00:44:29Z"},{"alias_kind":"arxiv_version","alias_value":"1306.3915v2","created_at":"2026-05-18T00:44:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.3915","created_at":"2026-05-18T00:44:29Z"},{"alias_kind":"pith_short_12","alias_value":"OPCR27HNDGRR","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"OPCR27HNDGRRBEXQ","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"OPCR27HN","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:a67d369172d5e5948b5af9bea16cad24466c411eb76fc482f47d6f91ef6913a1","target":"graph","created_at":"2026-05-18T00:44:29Z","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 define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that ","authors_text":"Helene Barcelo, Jacob A. White, Valerio Capraro","cross_cats":["math.AT","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-06-17T16:17:00Z","title":"Discrete Homology Theory for Metric Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3915","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":"sha256:5e7901f98ab0b09cb863c5fde32052b06ab74bc490a2fc8ce79bbc141c20cb17","target":"record","created_at":"2026-05-18T00:44:29Z","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":"6a4fe682430b9e1ddd456fe7f5e03120c563a9828c7dd5cc626c1a67f0f503c2","cross_cats_sorted":["math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-06-17T16:17:00Z","title_canon_sha256":"8e6ea94b1de9ff63e44b2d4246e582c19a472d1e029b02ab11085af36ae1836d"},"schema_version":"1.0","source":{"id":"1306.3915","kind":"arxiv","version":2}},"canonical_sha256":"73c51d7ced19a31092f00140ed8a653e7d6bfc7c9ab88a2ca2ecb9f9d56e4c2d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73c51d7ced19a31092f00140ed8a653e7d6bfc7c9ab88a2ca2ecb9f9d56e4c2d","first_computed_at":"2026-05-18T00:44:29.723865Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:29.723865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2Ta/qyqCIPlju1QWGjPmDMsWGX6AVX5JA6wPaedmTW5Pe7w0QUKn7pUh6fA4kwutfieQ0/lc5tQrKb+i+fUiCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:29.724385Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.3915","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e7901f98ab0b09cb863c5fde32052b06ab74bc490a2fc8ce79bbc141c20cb17","sha256:a67d369172d5e5948b5af9bea16cad24466c411eb76fc482f47d6f91ef6913a1"],"state_sha256":"4a39ec72f31cb3f567072bc3a3fb49e004ab29ed86053e7b077cb00beb412da6"}