{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:NTKHPUEIWKOOLOCMXTQC3VWOUA","short_pith_number":"pith:NTKHPUEI","canonical_record":{"source":{"id":"0711.0025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2007-10-31T22:25:48Z","cross_cats_sorted":[],"title_canon_sha256":"5e5ed6591a77cba2982d2b5efedf818593889dcce7783913e1409ae46f04abba","abstract_canon_sha256":"4489e5823cff6dd010012fbbe3af9198a6157263cec0059c4c57e72f47244b83"},"schema_version":"1.0"},"canonical_sha256":"6cd477d088b29ce5b84cbce02dd6cea0209ec9874d7f74a7288caaaf56676247","source":{"kind":"arxiv","id":"0711.0025","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.0025","created_at":"2026-05-18T04:23:13Z"},{"alias_kind":"arxiv_version","alias_value":"0711.0025v3","created_at":"2026-05-18T04:23:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.0025","created_at":"2026-05-18T04:23:13Z"},{"alias_kind":"pith_short_12","alias_value":"NTKHPUEIWKOO","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"NTKHPUEIWKOOLOCM","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"NTKHPUEI","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:NTKHPUEIWKOOLOCMXTQC3VWOUA","target":"record","payload":{"canonical_record":{"source":{"id":"0711.0025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2007-10-31T22:25:48Z","cross_cats_sorted":[],"title_canon_sha256":"5e5ed6591a77cba2982d2b5efedf818593889dcce7783913e1409ae46f04abba","abstract_canon_sha256":"4489e5823cff6dd010012fbbe3af9198a6157263cec0059c4c57e72f47244b83"},"schema_version":"1.0"},"canonical_sha256":"6cd477d088b29ce5b84cbce02dd6cea0209ec9874d7f74a7288caaaf56676247","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:13.538498Z","signature_b64":"a4cA2nHuhV2BRDmemM3gQlkIGwXmXxNMoKFpvJ0p3alWCj6AHPRQ6H0e8+K1yIpR1qsJcO7moLTEw3+Ovyi6BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6cd477d088b29ce5b84cbce02dd6cea0209ec9874d7f74a7288caaaf56676247","last_reissued_at":"2026-05-18T04:23:13.538043Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:13.538043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0711.0025","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"13aPb5PvZ6UGEDuJtVxY9Zc6o5lZESPLyupMLlc8amxoB/KRoKS9ffMkaSuj+dgJ42qmETSZHbl/iD4cvtfRAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:39:10.051202Z"},"content_sha256":"91e812c671f905dd9946d5c685ea13f605a7473fa7b138b8a294cdbf39d2ac5d","schema_version":"1.0","event_id":"sha256:91e812c671f905dd9946d5c685ea13f605a7473fa7b138b8a294cdbf39d2ac5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:NTKHPUEIWKOOLOCMXTQC3VWOUA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dualities in equivariant Kasparov theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Heath Emerson, Ralf Meyer","submitted_at":"2007-10-31T22:25:48Z","abstract_excerpt":"We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov's first and second Poincare duality isomorphisms.\n  We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the description of bivariant Kasparov theory for commutative C*-algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities apply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum-Conne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.0025","kind":"arxiv","version":3},"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":"2026-05-18T04:23:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CXZYI7UMx9wq8RlK7P+jDd27Nvd5IZkAQBfCgHqZMB6ZnX+g3d22JWZrUYZNsG6YlHxE8FvNQmak76QnfNf3AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:39:10.051564Z"},"content_sha256":"b4fb83a14f7554d4b13e2f09b7484b8bbbaf6a06da2b32615e28180ab21837ab","schema_version":"1.0","event_id":"sha256:b4fb83a14f7554d4b13e2f09b7484b8bbbaf6a06da2b32615e28180ab21837ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NTKHPUEIWKOOLOCMXTQC3VWOUA/bundle.json","state_url":"https://pith.science/pith/NTKHPUEIWKOOLOCMXTQC3VWOUA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NTKHPUEIWKOOLOCMXTQC3VWOUA/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":"2026-06-08T22:39:10Z","links":{"resolver":"https://pith.science/pith/NTKHPUEIWKOOLOCMXTQC3VWOUA","bundle":"https://pith.science/pith/NTKHPUEIWKOOLOCMXTQC3VWOUA/bundle.json","state":"https://pith.science/pith/NTKHPUEIWKOOLOCMXTQC3VWOUA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NTKHPUEIWKOOLOCMXTQC3VWOUA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:NTKHPUEIWKOOLOCMXTQC3VWOUA","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":"4489e5823cff6dd010012fbbe3af9198a6157263cec0059c4c57e72f47244b83","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2007-10-31T22:25:48Z","title_canon_sha256":"5e5ed6591a77cba2982d2b5efedf818593889dcce7783913e1409ae46f04abba"},"schema_version":"1.0","source":{"id":"0711.0025","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.0025","created_at":"2026-05-18T04:23:13Z"},{"alias_kind":"arxiv_version","alias_value":"0711.0025v3","created_at":"2026-05-18T04:23:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.0025","created_at":"2026-05-18T04:23:13Z"},{"alias_kind":"pith_short_12","alias_value":"NTKHPUEIWKOO","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"NTKHPUEIWKOOLOCM","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"NTKHPUEI","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:b4fb83a14f7554d4b13e2f09b7484b8bbbaf6a06da2b32615e28180ab21837ab","target":"graph","created_at":"2026-05-18T04:23:13Z","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 study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov's first and second Poincare duality isomorphisms.\n  We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the description of bivariant Kasparov theory for commutative C*-algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities apply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum-Conne","authors_text":"Heath Emerson, Ralf Meyer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2007-10-31T22:25:48Z","title":"Dualities in equivariant Kasparov theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.0025","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":"sha256:91e812c671f905dd9946d5c685ea13f605a7473fa7b138b8a294cdbf39d2ac5d","target":"record","created_at":"2026-05-18T04:23:13Z","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":"4489e5823cff6dd010012fbbe3af9198a6157263cec0059c4c57e72f47244b83","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2007-10-31T22:25:48Z","title_canon_sha256":"5e5ed6591a77cba2982d2b5efedf818593889dcce7783913e1409ae46f04abba"},"schema_version":"1.0","source":{"id":"0711.0025","kind":"arxiv","version":3}},"canonical_sha256":"6cd477d088b29ce5b84cbce02dd6cea0209ec9874d7f74a7288caaaf56676247","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6cd477d088b29ce5b84cbce02dd6cea0209ec9874d7f74a7288caaaf56676247","first_computed_at":"2026-05-18T04:23:13.538043Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:13.538043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"a4cA2nHuhV2BRDmemM3gQlkIGwXmXxNMoKFpvJ0p3alWCj6AHPRQ6H0e8+K1yIpR1qsJcO7moLTEw3+Ovyi6BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:13.538498Z","signed_message":"canonical_sha256_bytes"},"source_id":"0711.0025","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91e812c671f905dd9946d5c685ea13f605a7473fa7b138b8a294cdbf39d2ac5d","sha256:b4fb83a14f7554d4b13e2f09b7484b8bbbaf6a06da2b32615e28180ab21837ab"],"state_sha256":"399acc9f6a7e370e87473ca4138515c2cdff6b35da5ca3138928a8ad032753a4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jWjSg73FBWjdrjgDEjoPp30sB8f+CkipJh+bSlnN2hk9UUYWcSV2iqk6p0DDLPp5QYdn+6Ga7hdzVRTWE+rFAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T22:39:10.053681Z","bundle_sha256":"9db11f7c9c5a9c8de2734436c33980c752e24d50058c3d2c80bf2f99036bfe71"}}