{"paper":{"title":"On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three III - Generalized Heegaard splittings and mapping classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jungsoo Kim","submitted_at":"2015-09-01T06:39:26Z","abstract_excerpt":"Let $M$ be an orientable, irreducible $3$-manifold admitting a weakly reducible genus three Heegaard splitting as a minimal genus Heegaard splitting. In this article, we prove that if $[f]$, $[g]\\in Mod(M)$ give the same correspondence between two isotopy classes of generalized Heegaard splittings consisting of two Heegaard splittings of genus two, say $[\\mathbf{H}]\\to[\\mathbf{H}']$, then there exists a representative $h$ of the difference $[h]=[g]\\cdot[f]^{-1}$ such that (i) $h$ preserves a suitably chosen embedding of the Heegaard surface $F'$ obtained by amalgamation from $\\mathbf{H}'$ whic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00157","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}