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Let Emb_{\\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \\subset \\R^4 of radius r and of capacity c:= \\pi r^2 into (M,\\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \\ccrit \\in (0,w_{M}] such that, for all c\\in(0,\\ccrit), the embedding space Emb_{\\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \\SFr(M). We also know that the homotopy type of Emb_{\\omega}(B^{4}(c),M) changes wh"},"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":"0807.1031","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2008-07-07T15:13:04Z","cross_cats_sorted":[],"title_canon_sha256":"d8764c4da2749e553ba9befd5209da0f5369fe72dc39b1cf788eae0a978633a8","abstract_canon_sha256":"f5316cfc5c1b5b3bbf2f30b2713303d27a0134f61267d251ed480d19900e9120"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:19.096607Z","signature_b64":"Lmmujb0ZwnKiHJynO2c+9WAxDk7o+rvqC96A1VJFzTN1mxG2WAgad1VDHQdRq91/xI8XLnEaptPP5AkhTvTrBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfdc9243636a7806e490a4f412ad0ac78fdeacf24e4e34fe2fa2c3de15d7416e","last_reissued_at":"2026-05-18T02:38:19.095908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:19.095908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The homotopy type of the space of symplectic balls in rational ruled 4-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Francois Lalonde, Martin Pinsonnault, Silvia Anjos","submitted_at":"2008-07-07T15:13:04Z","abstract_excerpt":"Let M:=(M^{4},\\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. 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