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Let also $\\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and $\\mathcal{S}_V(f) = \\{h\\in\\mathcal{S}(f)\\,| h(V) = V\\}.$ The group $\\mathcal{S}_V(f)$ acts on the set $\\pi_0\\partial \\mathcal{E}_V$ of connected components of the boundary of $\\mathcal{E}_V.$ Therefore we have a homomorphism $\\phi:\\mathcal{S}(f)\\to \\math"},"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":"1610.01219","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AT","submitted_at":"2016-10-04T21:56:34Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"58aaec861d4cca190713d0177dc2888360e54c373b81c08757564c5658eab783","abstract_canon_sha256":"ae615571a3249d1636fe09a891d9d03d6d57031a34b733c5b1cc2df0b0b0104b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:11.393612Z","signature_b64":"icHib8UWgiYcDai2VWy0M//4S0cV4zKDGG1nFqKEgElHR8tUzDo48+SX2be5pFjV+FMdkz4kqssrMfUwH62vBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4bd139885faba5f327ded84b0f46d75f68eef14d43664e8db5d8fd7abbeb305c","last_reissued_at":"2026-05-18T01:03:11.393060Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:11.393060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Actions of finite groups and smooth functions on surfaces","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Bohdan Feshchenko","submitted_at":"2016-10-04T21:56:34Z","abstract_excerpt":"Let $f:M\\to \\mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and $\\mathcal{E}_V$ be its atom. 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