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We prove that this extension splits, so $\\text{Mod}(M_n)$ is the semidirect product of $\\text{Out}(F_n)$ by $(\\mathbb{Z}/2)^n$, which $\\text{Out}(F_n)$ acts on via the dual of the natural surjection $\\text{Out}(F_n) \\rightarrow \\text{GL}_n(\\mathbb{Z}/2)$. Our splitting takes $\\text{Out}(F_n)$ to the subgroup of $\\text{Mod}(M_n)$ consisting of mapping cl"},"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":"2012.01529","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2020-12-02T21:10:40Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"7167387ba9f7e50e0be62fb63bd0c6b6dbdfb1ba3a73cafc5670a9ef0a9a7540","abstract_canon_sha256":"7f1991034ed987394d2d2f0bb7c9993e32d7cc199e05da65382a9807f91b36f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:57:08.689475Z","signature_b64":"suQW7QvjQe+8Tnvh0G7om77AL6v6/cZfZbUc7TOqsz6UZzx/1W+XcVkwxciTE+m21zbr2g11QrCIqTH+dhXgBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5983da3c362ee7818bf8c88eb1d11715bdd20a49f522a955268f97f5e4d00e78","last_reissued_at":"2026-07-05T05:57:08.689124Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:57:08.689124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The mapping class group of connect sums of $S^2 \\times S^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Andrew Putman, Nathan Broaddus, Tara Brendle","submitted_at":"2020-12-02T21:10:40Z","abstract_excerpt":"Let $M_n$ be the connect sum of $n$ copies of $S^2 \\times S^1$. 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