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We show that the natural map of non-stable K_1-functors K_1^G(R)-> K_1^G(k((x_1))...((x_n))) is injective. This complements the surjectivity result for the same map obtained by V. Chernousov, P. Gille and A. Pianzola in arXiv:1109.5236. As a corollary, we provide a way to evaluate the difference betw"},"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":"1404.7587","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2014-04-30T04:58:16Z","cross_cats_sorted":["math.GR","math.RA"],"title_canon_sha256":"00ec84bfbfe9bbe0ae609e5f2a8dd119eb6cd20306cc8ad18178c5a9c511751b","abstract_canon_sha256":"e1eef948632307a98db78c34cb4602ad8d575ff921b474b77e7658bc7ccd93b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:55:33.915602Z","signature_b64":"OYKHpyM0l++i+1Wgh/rD8Td/Ho8YwPfNnQ9hdGaIZdlbfO/LnbDrQXZT4JP3kwdkwkxkoN9nSGPUD2e8NNgLBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa144fd0e07a7fc97369d99fdfc2095e01141c54a2f3ab822d1257291cd1e77d","last_reissued_at":"2026-05-18T01:55:33.915046Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:55:33.915046Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-stable K_1-functors of multiloop groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.RA"],"primary_cat":"math.KT","authors_text":"A. Stavrova","submitted_at":"2014-04-30T04:58:16Z","abstract_excerpt":"Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\\pm 1},...,x_n^{\\pm 1}] containing a maximal R-torus T (equivalently, loop reductive). Assume also that every semisimple normal subgroup of G contains a two-dimensional split torus G_m^2. We show that the natural map of non-stable K_1-functors K_1^G(R)-> K_1^G(k((x_1))...((x_n))) is injective. This complements the surjectivity result for the same map obtained by V. Chernousov, P. Gille and A. Pianzola in arXiv:1109.5236. 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