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Further, we define $E_{\\infty}$-maps $\\mathrm{MSpin} \\to (\\mathrm{MSpin}^c_{\\mathbb{R}})^{C_2}$ and $\\mathrm{MU}_{\\mathbb{R}} \\to \\mathrm{MSpin}^c_{\\mathbb{R}}$, which are used to recover the three well-known orientations of topological $\\mathrm{K}$-theory, $\\mathrm{MSpin}^c \\to \\mathrm{KU}$, $\\mathrm{MSpin} \\to \\mathrm{KO}$, and $\\mathrm{MU}_{\\mathbb{R}} \\to \\mathrm{KU}_{\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2405.00963","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-05-02T03:02:05Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"9e89f7d7c708d1161e4da897238f033c9b366fc0916dc56b67b51433fd7517c7","abstract_canon_sha256":"32189fdfb5ab0c4856648ba587e570349752d8728137c91309105378a3a5cb4e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:52:55.339494Z","signature_b64":"i0i2D9fga2CWjy22KtQAbO3ZLn+VO7PJG1QrUSv5Vyx0cwxr8ofhemyqCZXWEbfQcXTuu4rcd05zQuASnqklDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"193698be407f6060e89830d94e5b09ba7a9b6b177c31aebeff453f941afc49a5","last_reissued_at":"2026-07-05T11:52:55.339001Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:52:55.339001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Real spin bordism and orientations of topological $\\mathrm{K}$-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Yigal Kamel, Zachary Halladay","submitted_at":"2024-05-02T03:02:05Z","abstract_excerpt":"We construct a commutative orthogonal $C_2$-ring spectrum, $\\mathrm{MSpin}^c_{\\mathbb{R}}$, along with a $C_2$-$E_{\\infty}$-orientation $\\mathrm{MSpin}^c_{\\mathbb{R}} \\to \\mathrm{KU}_{\\mathbb{R}}$ of Atiyah's Real K-theory. Further, we define $E_{\\infty}$-maps $\\mathrm{MSpin} \\to (\\mathrm{MSpin}^c_{\\mathbb{R}})^{C_2}$ and $\\mathrm{MU}_{\\mathbb{R}} \\to \\mathrm{MSpin}^c_{\\mathbb{R}}$, which are used to recover the three well-known orientations of topological $\\mathrm{K}$-theory, $\\mathrm{MSpin}^c \\to \\mathrm{KU}$, $\\mathrm{MSpin} \\to \\mathrm{KO}$, and $\\mathrm{MU}_{\\mathbb{R}} \\to \\mathrm{KU}_{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.00963","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.00963/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2405.00963","created_at":"2026-07-05T11:52:55.339059+00:00"},{"alias_kind":"arxiv_version","alias_value":"2405.00963v2","created_at":"2026-07-05T11:52:55.339059+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.00963","created_at":"2026-07-05T11:52:55.339059+00:00"},{"alias_kind":"pith_short_12","alias_value":"DE3JRPSAP5QG","created_at":"2026-07-05T11:52:55.339059+00:00"},{"alias_kind":"pith_short_16","alias_value":"DE3JRPSAP5QGB2EY","created_at":"2026-07-05T11:52:55.339059+00:00"},{"alias_kind":"pith_short_8","alias_value":"DE3JRPSA","created_at":"2026-07-05T11:52:55.339059+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ","json":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ.json","graph_json":"https://pith.science/api/pith-number/DE3JRPSAP5QGB2EYGDMU4WYJXJ/graph.json","events_json":"https://pith.science/api/pith-number/DE3JRPSAP5QGB2EYGDMU4WYJXJ/events.json","paper":"https://pith.science/paper/DE3JRPSA"},"agent_actions":{"view_html":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ","download_json":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ.json","view_paper":"https://pith.science/paper/DE3JRPSA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2405.00963&json=true","fetch_graph":"https://pith.science/api/pith-number/DE3JRPSAP5QGB2EYGDMU4WYJXJ/graph.json","fetch_events":"https://pith.science/api/pith-number/DE3JRPSAP5QGB2EYGDMU4WYJXJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ/action/storage_attestation","attest_author":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ/action/author_attestation","sign_citation":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ/action/citation_signature","submit_replication":"https://pith.science/pith/DE3JRPSAP5QGB2EYGDMU4WYJXJ/action/replication_record"}},"created_at":"2026-07-05T11:52:55.339059+00:00","updated_at":"2026-07-05T11:52:55.339059+00:00"}