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Our main result is that (modulo two technical conditions on $(\\pi,w)$) there are at most $2^\\beta$ orbits of $k$-invariants determining \"strongly minimal\" complexes (i.e., those with homotopy intersection pairing $\\lambda_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $\\pi$ a $PD_2$-group is determined by $\\pi$, $w$, $\\lambda_X$ and the $v_2$-type of $X$. Our result also implies that Fox's 2-knot with metabeli"},"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":"0712.1069","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2007-12-07T02:09:30Z","cross_cats_sorted":[],"title_canon_sha256":"8b2df1b5a5fc0a4ed47ceed364c3f2a86a07c7c23e6eed861f1a09e91a3fb6cd","abstract_canon_sha256":"a73bd4df4435ba499628d01b132af49e522083622f72c7c032ecf1d1605f3974"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:47.287394Z","signature_b64":"NNVkPkvYUFHmWfOmJVSLLhI3S7QqRVCTPrbdDOFw7n9R6nB5dtYFI6Q59KnFQSKF575HN0LfYQJnv4sGT6SFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db16864fe30783111255c125ad6cd7345ff7780d269f22800adf60117cb7d252","last_reissued_at":"2026-05-18T04:10:47.286988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:47.286988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strongly minimal PD4-complexes","license":"","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jonathan A. 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