{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:RAERYOLABVCI7F52A3Q333TT7X","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":"b3c1b082223215467fd2f1c97a12232712c22860983144886e778a657e02b615","cross_cats_sorted":[],"license":"","primary_cat":"math.KT","submitted_at":"2004-07-05T11:56:05Z","title_canon_sha256":"9482f7a6541b73303c15f824b9d94fc03ace68b3449bb29f8099ddc20f6f85c2"},"schema_version":"1.0","source":{"id":"math/0407054","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0407054","created_at":"2026-07-04T14:50:06Z"},{"alias_kind":"arxiv_version","alias_value":"math/0407054v2","created_at":"2026-07-04T14:50:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0407054","created_at":"2026-07-04T14:50:06Z"},{"alias_kind":"pith_short_12","alias_value":"RAERYOLABVCI","created_at":"2026-07-04T14:50:06Z"},{"alias_kind":"pith_short_16","alias_value":"RAERYOLABVCI7F52","created_at":"2026-07-04T14:50:06Z"},{"alias_kind":"pith_short_8","alias_value":"RAERYOLA","created_at":"2026-07-04T14:50:06Z"}],"graph_snapshots":[{"event_id":"sha256:1efc7aa5399a44456af514736c9613f5965679b5debf769625ce21be82b527de","target":"graph","created_at":"2026-07-04T14:50:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0407054/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Twisted complex $K$-theory can be defined for a space $X$ equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C$^*$-algebras. Up to equivalence, the twisting corresponds to an element of $H^3(X;\\Z)$. We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary $K$-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that t","authors_text":"Graeme Segal, Michael Atiyah","cross_cats":[],"headline":"","license":"","primary_cat":"math.KT","submitted_at":"2004-07-05T11:56:05Z","title":"Twisted $K$-theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0407054","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:eab6fd2851924c50460e582160cf1bfc7d6c4154ce507de0fea729b26eb9652f","target":"record","created_at":"2026-07-04T14:50:06Z","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":"b3c1b082223215467fd2f1c97a12232712c22860983144886e778a657e02b615","cross_cats_sorted":[],"license":"","primary_cat":"math.KT","submitted_at":"2004-07-05T11:56:05Z","title_canon_sha256":"9482f7a6541b73303c15f824b9d94fc03ace68b3449bb29f8099ddc20f6f85c2"},"schema_version":"1.0","source":{"id":"math/0407054","kind":"arxiv","version":2}},"canonical_sha256":"88091c39600d448f97ba06e1bdee73fdecd35b316827db0a7a5a92cc6258ff1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"88091c39600d448f97ba06e1bdee73fdecd35b316827db0a7a5a92cc6258ff1a","first_computed_at":"2026-07-04T14:50:06.213656Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:50:06.213656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VO7s9whzpEYXs3L4DpeeHkI1fwIKrU7T7pWoBCWd6UhLUVpIfGl0w5AJjM/r4nDKZXExGDpJ54Ko+PuKUEuWBA==","signature_status":"signed_v1","signed_at":"2026-07-04T14:50:06.214057Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0407054","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eab6fd2851924c50460e582160cf1bfc7d6c4154ce507de0fea729b26eb9652f","sha256:1efc7aa5399a44456af514736c9613f5965679b5debf769625ce21be82b527de"],"state_sha256":"4b17a0861f112b4406fc68a1802430985925f82c88b9654620eb1b5e65d0597a"}