{"paper":{"title":"The homotopy fibre of the inclusion $F\\_n(M) \\lhook\\joinrel\\longrightarrow \\prod\\_{1}^{n} M$ for $M$ either $\\mathbb{S}^2$ or$\\mathbb{R}P^2$ and orbit configuration spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Daciberg Lima Gon\\c{c}alves (USP, IME), John Guaschi (LMNO, NU), UNICAEN","submitted_at":"2017-10-31T15:58:22Z","abstract_excerpt":"Let $n\\geq 1$, and let $\\iota\\_{n}\\colon\\thinspace F\\_{n}(M) \\longrightarrow \\prod\\_{1}^{n} M$ be the natural inclusion of the $n$th configuration space of $M$ in the $n$-fold Cartesian product of $M$ with itself. In this paper, we study the map $\\iota\\_{n}$, its homotopy fibre $I\\_{n}$, and the induced homomorphisms $(\\iota\\_{n})\\_{#k}$ on the $k$th homotopy groups of $F\\_{n}(M)$ and $\\prod\\_{1}^{n} M$ for $k\\geq 1$ in the cases where $M$ is the $2$-sphere $\\mathbb{S}^{2}$ or the real projective plane $\\mathbb{R}P^{2}$. If $k\\geq 2$, we show that the homomorphism $(\\iota\\_{n})\\_{#k}$ is injec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.11544","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"}