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We prove that $\\mathcal{F}_h^i(k,n)$ is (when non empty) a complex sub\\-ma\\-ni\\-fold of $Gr(k,n)^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk \\neq n$ and $n>2$ and equal to the braid group of the sphere $\\mc P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$."},"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":"1311.5642","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-11-22T03:05:30Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"bb4068c8717b2eda90d5e43eaff93019394d51c6481fc6d811e9f3cede13112e","abstract_canon_sha256":"c9f256ef42285a882d168877ad17572ac37d209e085ce93d3384b5d15e20eca9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:23.653056Z","signature_b64":"t08Y/psA5aruGZemhD4Tc2k+1c+bysXmoBrtwurL66OCTD3kwPbaGh1LhxAhxDQBTseJ/8O1ch5hWmhUCQ20Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e37048b39008aba902aa084180fc8e8f44fcfcbe500d26ff999e70d51955649","last_reissued_at":"2026-05-18T03:06:23.652489Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:23.652489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Configuration Spaces of Grassmannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GR","authors_text":"Sandro Manfredini, Simona Settepanella","submitted_at":"2013-11-22T03:05:30Z","abstract_excerpt":"Let $\\mathcal{F}_h^i(k,n)$ be the $i$th ordered configuration space of all distinct points $H_1,\\ldots,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of $\\mc^n$, whose sum is a subspace of dimension $i$. We prove that $\\mathcal{F}_h^i(k,n)$ is (when non empty) a complex sub\\-ma\\-ni\\-fold of $Gr(k,n)^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk \\neq n$ and $n>2$ and equal to the braid group of the sphere $\\mc P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.5642","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1311.5642","created_at":"2026-05-18T03:06:23.652577+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.5642v1","created_at":"2026-05-18T03:06:23.652577+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.5642","created_at":"2026-05-18T03:06:23.652577+00:00"},{"alias_kind":"pith_short_12","alias_value":"BY3QJCZZACFL","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_16","alias_value":"BY3QJCZZACFLVEBK","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_8","alias_value":"BY3QJCZZ","created_at":"2026-05-18T12:27:40.988391+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D","json":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D.json","graph_json":"https://pith.science/api/pith-number/BY3QJCZZACFLVEBKUCCBQD6I5D/graph.json","events_json":"https://pith.science/api/pith-number/BY3QJCZZACFLVEBKUCCBQD6I5D/events.json","paper":"https://pith.science/paper/BY3QJCZZ"},"agent_actions":{"view_html":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D","download_json":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D.json","view_paper":"https://pith.science/paper/BY3QJCZZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.5642&json=true","fetch_graph":"https://pith.science/api/pith-number/BY3QJCZZACFLVEBKUCCBQD6I5D/graph.json","fetch_events":"https://pith.science/api/pith-number/BY3QJCZZACFLVEBKUCCBQD6I5D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D/action/storage_attestation","attest_author":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D/action/author_attestation","sign_citation":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D/action/citation_signature","submit_replication":"https://pith.science/pith/BY3QJCZZACFLVEBKUCCBQD6I5D/action/replication_record"}},"created_at":"2026-05-18T03:06:23.652577+00:00","updated_at":"2026-05-18T03:06:23.652577+00:00"}