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In the case of Sullivan extensions $\\land W\\to \\land W\\otimes \\land Z\\to \\land Z$ in which dim$\\, H(\\land Z)<\\infty$ we show that $${\\mathcal T}(\\land W\\otimes \\land Z)= {\\mathcal T}(\\land W)\\otimes {\\mathcal T}(\\land Z).$$ This is applied to finite dimensional CW complexes $X$ where the fundamental group $G$ "},"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":"1812.09686","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-12-23T10:39:13Z","cross_cats_sorted":[],"title_canon_sha256":"c6f6402c88dd542b7dd68c93e3e657d53ee27790437d50a56b94c98b18e9abad","abstract_canon_sha256":"ed9b2bec3d800b28a1c2cb421376fc911e8c14699c536f922b0602f4e69f902b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:27.108642Z","signature_b64":"xee+X88iTlDoVxxYBMF2J0gZpxgTFRgCaKYjKaThOkdbSI0mcBExKN6/VyMj4c/WHnGcbDO9FXd1WZWqO+MHBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43714dcf02cd977cb0847f4201ccc5a21519f7b81d5e9a33c97c42f035492b9e","last_reissued_at":"2026-05-17T23:57:27.108013Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:27.108013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on Gorenstein spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Steve Halperin, Yves Felix","submitted_at":"2018-12-23T10:39:13Z","abstract_excerpt":"Associated with an augmented differential graded algebra $R= R^{\\geq 0}$ is a homotopy invariant ${\\mathcal T}(R)$. This is a graded vector space, and if $H^0(R)$ is the ground field and $H^{>N}(R)= 0$ then dim$\\, {\\mathcal T}(R)= 1$ if and only if $H(R)$ is a Poincar\\'e duality algebra. 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