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As the first application, we show that the connecting map $G\\rightarrow\\mathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $\\mathrm{Map}_0(B_nG,BG)\\rightarrow\\mathrm{Map}("},"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":"1408.2010","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-08-09T02:52:09Z","cross_cats_sorted":[],"title_canon_sha256":"00af7d563abc031441ffae685974ed0f7c55d7ed2371c82e72005957882e9dbb","abstract_canon_sha256":"36c4eb6dc6042a76cf6c6b965f6e35c8782564c432517700ffb8c0febbf32e0f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:32.365941Z","signature_b64":"jZYlDSzJJ6JzNulh/Dld9BN6OJug1gpw+H7icmHUpB4fNYzoS11qWZaTGgEA8C+pWJeiESHgLoJZSN+Q8C22Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46bd5772d876d4c205c86bcfdd8beeda4830f921c457255407e0c8f610a2b6b9","last_reissued_at":"2026-05-18T01:09:32.365454Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:32.365454Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mapping spaces from projective spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Mitsunobu Tsutaya","submitted_at":"2014-08-09T02:52:09Z","abstract_excerpt":"We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. 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