{"paper":{"title":"A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Alessandro Achille, Alessandro Berarducci","submitted_at":"2017-06-07T09:03:21Z","abstract_excerpt":"Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\\to Y$ with strong connectivity conditions on the fibers is given. We apply similar techniques in o-minimal expansions of fields to compare the o-minimal homotopy of a definable set $X$ with the homotopy of some of its bounded hyperdefinable quotients $X/E$. Under suitable assumption, we show that $\\pi_{n}(X)^{\\rm def}\\cong\\pi_{n}(X/E)$ and $\\dim(X)=\\dim_{\\mathbb R}(X/E)$. As a special case, given a definably compact group, we obtain a new proof of Pillay's "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02094","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"}