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Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions, we can guarantee that every non-trivial homomorphism is a surjection. As a corollary, for any fixed integer $m \\ge 5$, it is NP-complete to decide whether $M$ admits a connected $m$-sheeted covering.\n  Our construction is inspired by universality results in topological quantum c"},"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":"1707.03811","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-07-12T17:39:46Z","cross_cats_sorted":["cs.CC","math.GR"],"title_canon_sha256":"f4e7cbf72bece7e868d878f2874c95ce79f63597f1fefd27f572bd2e05735bd7","abstract_canon_sha256":"07200005ee1b5fe86bbc443012d1619307e7382cae0f3ffad13c207d47381192"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:17.824734Z","signature_b64":"IInT84VAZS63cFWuFSjcq1vnkPLBEYoitOOwh30vHfR0LSQwbpC/e6I2acy8ePsOPZuBjRFQB2ffrYqp5XqQAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae4dbd54f7407695a4059521365ffc21d33c3494f526df5bd2969374654d824a","last_reissued_at":"2026-05-18T00:04:17.823983Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:17.823983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computational complexity and 3-manifolds and zombies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.GR"],"primary_cat":"math.GT","authors_text":"Eric Samperton (UC Davis), Greg Kuperberg (UC Davis)","submitted_at":"2017-07-12T17:39:46Z","abstract_excerpt":"We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. 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