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In this paper, we construct an 124-vertex simplicial subdivision $(S^2 \\times S^2 \\times S^2)_{124}$ of the 64-vertex standard cellulation $S^2_4 \\times S^2_4 \\times S^2_4$ of $S^2 \\times S^2 \\times S^2$, such that the $S_3$-action on this cellulation naturally extends to an action on $(S^2 \\times S^2 \\times S^2)_{124}$. Further, the $S_3$-action on $(S^2 \\times S^2 \\times S^2)_{124}$ is \"goo"},"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":"1012.3235","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-12-15T06:42:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9d7184e6c0f30f09859618fa4234cb915fbd66a89b05b3324e211351dccf39ec","abstract_canon_sha256":"e84c4638c2cae9cdff80c27fb1e3394e85beba61232d60a91636dda0e69ee868"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:20.032941Z","signature_b64":"cdQjY9Mw3EOmLVqmUUnWLo6Hgmdz65KDrsmk8UsZTPxeI+ea+lDNSlkULKQ8KUmhBMoi6uQdG3hgAe3IZ1nMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59c9ae4035849b760275e95362dd6cc983aa913cffccf5d9d51789d092bce141","last_reissued_at":"2026-05-18T03:49:20.032372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:20.032372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A triangulation of $\\CC P^3$ as symmetric cube of $S^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AT","authors_text":"Basudeb Datta, Bhaskar Bagchi","submitted_at":"2010-12-15T06:42:31Z","abstract_excerpt":"The symmetric group $S_3$ acts on $S^2 \\times S^2 \\times S^2$ by coordinate permutation, and the quotient space $(S^2 \\times S^2 \\times S^2)/S_3$ is homeomorphic to the complex projective space $\\CC P^3$. In this paper, we construct an 124-vertex simplicial subdivision $(S^2 \\times S^2 \\times S^2)_{124}$ of the 64-vertex standard cellulation $S^2_4 \\times S^2_4 \\times S^2_4$ of $S^2 \\times S^2 \\times S^2$, such that the $S_3$-action on this cellulation naturally extends to an action on $(S^2 \\times S^2 \\times S^2)_{124}$. 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