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For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples constructed by Joyce by desingularising orbifolds have odd nu.\n  We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) determines a G_2-structure up to homotopy and di"},"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":"1211.0269","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-11-01T19:54:17Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"cdaec946990d5eb685a71ddd0a477ec9a024732ebb46815f9beaca3fcef5b353","abstract_canon_sha256":"7f9d6f8c63f4645e1812d0979a939af4040bbe039f4e17261a319b0cbb0db40b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:44.522928Z","signature_b64":"ZN4zwR3JYNDlw5cOkZMxpjxmQcGgz7kisXzu/o3lWVJ1t3JLAtU6GpmjFHhRces8MgIrrJbt/TTb52rGNvLWDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccaa6da289372571e133628bdc8b3ff342ca8a052982ed4110189024d337d897","last_reissued_at":"2026-05-18T01:28:44.522532Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:44.522532Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New invariants of G_2-structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Diarmuid Crowley, Johannes Nordstr\\\"om","submitted_at":"2012-11-01T19:54:17Z","abstract_excerpt":"We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. 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