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Consider the endomorphism $\\phi_k^*$ induced by $\\phi$ in the cohomology of $M$ of degree $k$, and denote by $J_k(\\lambda)$ the maximal size of its Jordan block of eigenvalue $\\lambda$. Define a representation $\\rho_\\lambda : \\pi_1(X)\\to\\mathbb{C}^*$ by $\\rho_\\lambda (g) = \\lambda^{p_*(g)}.$ Let $H^*(X,\\rho_\\lambda)$ be the corresponding tw"},"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":"1610.01136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-10-04T19:37:49Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"b070faaa2e583149bd87e187aace69a1a1dc183422e9cbc385015208c051f214","abstract_canon_sha256":"cbdcf67723f5044db6ca3fead3d9d42b4796324748d1d05a6696dcfa98627f27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:46.784730Z","signature_b64":"UTbw8CrqVaDO90rEz337wA/PbxnVHZ4A8b/vSiHFg1N3SWXi6EOfGWg6rwnS0o8ZiAMra4bHtxFiy3i7ShgFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"089a311f329ac26952ba23a760200344d837d6bf4fd718ba6bdde122c490553d","last_reissued_at":"2026-05-18T00:30:46.784108Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:46.784108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Massey products in mapping tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Andrei Pajitnov","submitted_at":"2016-10-04T19:37:49Z","abstract_excerpt":"Let $\\phi: M\\to M$ be a diffeomorphism of a $C^\\infty$ compact connected manifold, and $X$ its mapping torus. 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