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We consider the $\\mathbb{R}$-valued flux homomorphism on $\\rm{Symp}(D,0)_{\\rm{rel}}$ and define the central $\\mathbb{R}$-extension called the $\\mathbb{R}$-valued flux extension. We determine the Euler class of this extension and investigate the relation between the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$."},"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":"1905.08029","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-05-20T12:24:31Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"502633b1078e60c5987e5cde41833b249425cf8ac7fb062b500d22a9711071f9","abstract_canon_sha256":"17a7c706c3ebd155d8b6f15923e19e530c04055b8784ea57ceb4fdc6f400edf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:10.775136Z","signature_b64":"mjL5d/T/Rs5rDG0A7KThuF3vT1j5HV4r1L/m3hetun+y90uEo00GgKAR5bnQ4//jomSmf2pJXJHqrIw170C7Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08ee890156ae599417dd1fbd653981bc04d57bac4a88d3087b440bb4c8e7e121","last_reissued_at":"2026-05-17T23:40:10.774624Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:10.774624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The flux homomorphism and central extensions of diffeomorphism groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.GT","authors_text":"Shuhei Maruyama","submitted_at":"2019-05-20T12:24:31Z","abstract_excerpt":"Let $D$ be a 2-dimensional closed unit disk and $\\rm{Symp}(D,0)_{\\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\\partial D$ pointwise. 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