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Let $C'$ be the subcomplex of the free splitting complex obtained from $\\partial\\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\\partial\\mathcal{FS}\\hookrightarrow C'$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $\\theta$"},"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":"2608.04456","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-08-05T05:26:04Z","cross_cats_sorted":["math.CO","math.GR","math.QA"],"title_canon_sha256":"920a11d1eed6148cddf90172724a4a81748c2590b584895105475e04342458ed","abstract_canon_sha256":"b07b0536e53415b8b1c4f9d5f0eaa49835c500a75df7320b6fb806fe536c3c2f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-06T01:35:30.950027Z","signature_b64":"QTt2PkpyrjDgvVYJ3bYOOKjZWLhLMMfjXDDClepv5zBanLnGTPgQ6VaEoMWfKiWOvIVeNtkDDGyRsCgQXLibBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ac96676eb9a1997937eb675a95d07bde4b13f723f43dfe869ce6a036fcb96dd","last_reissued_at":"2026-08-06T01:35:30.948462Z","signature_status":"signed_v1","first_computed_at":"2026-08-06T01:35:30.948462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bond thickenings of the simplicial boundary of Outer space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR","math.QA"],"primary_cat":"math.AT","authors_text":"Benjamin Br\\\"uck","submitted_at":"2026-08-05T05:26:04Z","abstract_excerpt":"We study the simplicial boundary $\\partial\\mathcal{FS}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C'$ be the subcomplex of the free splitting complex obtained from $\\partial\\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\\partial\\mathcal{FS}\\hookrightarrow C'$ is $(2n-3)$-connected. 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