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The aim of this paper is to study some properties of the two following sequences of $L^2$ cohomology groups: $H^i_{2,m\\rightarrow M}(M,g)$ defined as the image $\\im(H^i_{2,min}(M,g)\\rightarrow H^i_{2,max}(M,g))$ and $\\bar{H}^i_{2,m\\rightarrow M}(M,g)$ defined as $\\im(\\bar{H}^i_{2,min}(M,g)\\rightarrow \\bar{H}^i_{2,max}(M,g))$. We show, under certain hypothesis, that the first sequence is the cohomology of a suitable Hilbert complex which contains the minimal one and is contained in the maximal one. We also show that when the second sequence "},"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":"1209.3528","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-16T23:07:15Z","cross_cats_sorted":[],"title_canon_sha256":"2e111a7e0985707d9cbdfc1509a7da8783eb67dea00f5f4094d81164870463b3","abstract_canon_sha256":"6be8340a73da8f1b4b8764170fb4c027f048d8e61d72b718019055b528b4bf42"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:00.930113Z","signature_b64":"eo7DDtr14dpxddKVzZmMQ7O61/IXQtQT5blORSGhkWFw9x7sgayHgQ5peRJwdFysQG9WlVPJ3nObJesyytRyCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"796a49b40d467c50ea97147d4b066c3b3943fdca519b8f93e22bc2c240559499","last_reissued_at":"2026-05-18T02:43:00.929612Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:00.929612Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Poincar\\'e duality, Hilbert complexes and geometric applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Francesco Bei","submitted_at":"2012-09-16T23:07:15Z","abstract_excerpt":"Let $(M,g)$ an open and oriented riemannian manifold. The aim of this paper is to study some properties of the two following sequences of $L^2$ cohomology groups: $H^i_{2,m\\rightarrow M}(M,g)$ defined as the image $\\im(H^i_{2,min}(M,g)\\rightarrow H^i_{2,max}(M,g))$ and $\\bar{H}^i_{2,m\\rightarrow M}(M,g)$ defined as $\\im(\\bar{H}^i_{2,min}(M,g)\\rightarrow \\bar{H}^i_{2,max}(M,g))$. We show, under certain hypothesis, that the first sequence is the cohomology of a suitable Hilbert complex which contains the minimal one and is contained in the maximal one. 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