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We extend the results obtained in the simply connected case, and prove in particular that if $X\\cup_fD^{n+1}$ is a Poincar\\'e duality complex and the algebra $H(X)$ requires at least two generators then $[f]\\in \\pi_n(X)$ is rationally inert. On the other hand, if $X$ is rationally a wedge of at least two spheres and $f$ is rationally non trivial, then $f$ is rationally inert. Finally if $f$ is rationally inert then the "},"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":"1904.08714","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-04-18T12:03:57Z","cross_cats_sorted":[],"title_canon_sha256":"0a4423b72ec342ebdc7a40d69be0e193e9f4e086038f11b6612bef750ba09aeb","abstract_canon_sha256":"32f4123ed13dc6419676f005d0145f9b53d6cb4c29275259d7c151cb01ae27e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:13.266146Z","signature_b64":"6plbNZgrj2Ll8cxiG/otIpEPA3lnhzlG4c6YCX6LaxYrfdCBlvP9WKPpthD4Q1Bx5/S/fTW9HgtBiS9MMy+oCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e638d73eee6e9830a9329b48ece2f618f98426b5809182a3863f0170d4ad192","last_reissued_at":"2026-05-17T23:48:13.265428Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:13.265428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Aspherical completions and rationally inert elements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Steve Halperin, Yves Felix","submitted_at":"2019-04-18T12:03:57Z","abstract_excerpt":"Let $X$ be a connected space. An element $[f]\\in \\pi_n(X)$ is called rationally inert if\n  $\\pi_*(X)\\otimes \\mathbb Q \\to \\pi_*(X\\cup_fD^{n+1})\\otimes \\mathbb Q$ is surjective. We extend the results obtained in the simply connected case, and prove in particular that if $X\\cup_fD^{n+1}$ is a Poincar\\'e duality complex and the algebra $H(X)$ requires at least two generators then $[f]\\in \\pi_n(X)$ is rationally inert. On the other hand, if $X$ is rationally a wedge of at least two spheres and $f$ is rationally non trivial, then $f$ is rationally inert. 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