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How close is stable homeomorphism to homeomorphism?\n  When the common fundamental group $\\pi$ is virtually abelian, we show that large $r$ can be diminished to $n+2$, where $\\pi$ has a finite-index subgroup that is free-abelian of rank $n$. In particular, if $\\pi$ is finite then $n=0$, hence $X$ and $Y$ are $2$-stably homeomorphic, which is one $S^2 \\times S^2$ summa"},"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":"1606.05968","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-06-20T04:22:11Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"9469a86a3ae9a5a5b43172bf1ddaf23251a095612ff43fd78d51d6a6f8f9a502","abstract_canon_sha256":"5d3a689c3bbf9f439a83f18ce7162bb89550a03cd1679f4c7244e8357e370206"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:17.155363Z","signature_b64":"0CDLbi1gkZpXCtg5sM87017CV8o/JKQlh1zYGLo7SsSU7kwJN9G+8Z2mUhtItW6aiamChwAcDeRF5O1RweM+AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91c48354f8a0905f88a042a24eb41921be107ee798f67e419bcf9009f43bb449","last_reissued_at":"2026-05-18T00:51:17.154729Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:17.154729Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cancellation for 4-manifolds with virtually abelian fundamental group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Qayum Khan","submitted_at":"2016-06-20T04:22:11Z","abstract_excerpt":"Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $\\pi$. For any $r \\geqslant 0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y \\# r(S^2 \\times S^2)$ is homeomorphic to $X \\# r(S^2\\times S^2)$. How close is stable homeomorphism to homeomorphism?\n  When the common fundamental group $\\pi$ is virtually abelian, we show that large $r$ can be diminished to $n+2$, where $\\pi$ has a finite-index subgroup that is free-abelian of rank $n$. 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