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It is proved by Crabb--Sutherland and the second author that the number of $A_n$-types of the gauge groups of $P$ is finite if $n<\\infty$ and $K$ is a finite complex. We show that the number of $A_\\infty$-types of the gauge groups of $P$ is infinite if $K$ is a sphere and there are infinitely many $P$."},"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":"1502.02781","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-02-10T05:14:58Z","cross_cats_sorted":[],"title_canon_sha256":"7a10ef98675e2798c2f95b45e4cd550a40ba4a80d4962e5e2d9e724755664f86","abstract_canon_sha256":"64e03ff020b087a12097785bb5e3215fc90849381846a83fec29a6df53eb1815"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:32.311876Z","signature_b64":"qQ0zPLw9qbVDgIJoVGE0fpDC8n0sPuYTpvkicHuKURfWXRMpbLD04pxpqspB9NsnMEPpmxkmw8PquBDuG3aqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa98004bdacb8d14f7d5df7f11cfa3d5b549192cc73aac338b8ce5053af47bdf","last_reissued_at":"2026-05-18T01:09:32.311377Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:32.311377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Infiniteness of $A_\\infty$-types of gauge groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Daisuke Kishimoto, Mitsunobu Tsutaya","submitted_at":"2015-02-10T05:14:58Z","abstract_excerpt":"Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. 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