{"paper":{"title":"On the Shrinkable U.S.C. Decomposition Spaces of Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Shijie Gu","submitted_at":"2013-12-04T05:41:54Z","abstract_excerpt":"Let $G$ be a u.s.c decomposition of $S^n$, $H_G$ denote the set of nondegenerate elements and $\\pi$ be the projection of $S^n$ onto $S^n/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with ($n-1$)-sphere frontiers which miss $\\pi(H_G)$, and such frontiers satisfies the Mismatch Property. Then this paper shows that this condition implies $S^n/G$ is homeomorphic to $S^n$ ($n\\geq 4$). This answers a weakened form of a conjecture asked by Daverman [3, p. 61]. In the case $n=3$, the strong form of the conjecture has an affirmative answer from Woodruff [12"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.1030","kind":"arxiv","version":3},"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"}