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Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\\mathrm{Map}^h(\\mathsf D_m,\\mathsf D_n^{\\mathbb Q})$ and $\\mathrm{Map}^h(\\mathsf D_m,\\mathsf D_n)$ are simply connected and rationally equivalent. As application we determine the ratio"},"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":"1703.06123","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2017-03-17T17:35:42Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"779c0544879039270fcf037aa09cbab0e7f902c73fcdf24870d981edaf7e6ad5","abstract_canon_sha256":"127bcc3d2ce9351f65c649dc63ef6f60bd1d377f42ce30ea10da5ad83ab8e080"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:29.421594Z","signature_b64":"uPlcLphDXw1+FCz/PtkciO8XC5QrwD9bjawoC9nsb5lNcuJWA5gKeMS/wuDgcrKEk68HEbtXZtB8UdioCBMSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e714a495d9967655c7d020c318e69b8301287ded17addba38ec690eadf94fff9","last_reissued_at":"2026-05-18T00:48:29.421013Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:29.421013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The rational homotopy of mapping spaces of E${}_n$ operads","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.QA","authors_text":"Benoit Fresse, Thomas Willwacher, Victor Turchin","submitted_at":"2017-03-17T17:35:42Z","abstract_excerpt":"We express the rational homotopy type of the mapping spaces $\\mathrm{Map}^h(\\mathsf D_m,\\mathsf D_n^{\\mathbb Q})$ of the little discs operads in terms of graph complexes. 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