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In this article, we develop methods for studying the low-dimensional homotopy groups of these spaces and of their subspaces Y of irreducible representations.\n  Our main result is that when G = GL(n,C) or SL(n,C), the second homotopy group of X is trivial. The proof depends on a new general position-type result in a singular setting. This result is proven in the Appendix and may be of independent interest.\n  We also obtain"},"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":"1412.0272","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-11-30T19:58:18Z","cross_cats_sorted":["math.AG","math.RT"],"title_canon_sha256":"6678c7c39d9e7aabe2baf98aa65a39ecca1cf1dd37626c1e79348d1a58dd9a09","abstract_canon_sha256":"fb5713b6177ba77528b0a31016e09ab5e25dc31a72c650e7c9d668c66b2257af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:39.299785Z","signature_b64":"MsjL5jAUe/LK/2/Ty69ia0WUP4QLqxCOjFl6pFcMLrJ5Ife+2865tMlsMwAA9RQH58RYDDRkN+Tcb4Ci1iLgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e0e29e13c25301bf4baba509421566b5a051c9506b6c468bdc7e9bca3e87394","last_reissued_at":"2026-05-18T00:16:39.299147Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:39.299147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homotopy Groups of Free Group Character Varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.AT","authors_text":"Carlos Florentino, Daniel Ramras, Sean Lawton","submitted_at":"2014-11-30T19:58:18Z","abstract_excerpt":"Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the low-dimensional homotopy groups of these spaces and of their subspaces Y of irreducible representations.\n  Our main result is that when G = GL(n,C) or SL(n,C), the second homotopy group of X is trivial. The proof depends on a new general position-type result in a singular setting. 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