{"paper":{"title":"Odd dimensional analogue of the Euler characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc","math.GN"],"primary_cat":"hep-th","authors_text":"L. Borsten, M. J. Duff, S. Nagy","submitted_at":"2021-05-27T16:11:32Z","abstract_excerpt":"When compact manifolds $X$ and $Y$ are both even dimensional, their Euler characteristics obey the K\\\"unneth formula $\\chi(X\\times Y)=\\chi(X) \\chi(Y)$. In terms of the Betti numbers $b_p(X)$, $\\chi(X)=\\sum_{p}(-1)^p b_p(X)$, implying that $\\chi(X)=0$ when $X$ is odd dimensional. We seek a linear combination of Betti numbers, called $\\rho$, that obeys an analogous formula $\\rho(X\\times Y)=\\chi(X) \\rho(Y)$ when $Y$ is odd dimensional. The unique solution is $\\rho(Y)=-\\sum_{p}(-1)^p p b_p(Y)$. Physical applications include: (1) $\\rho \\rightarrow (-1)^m \\rho $ under a generalized mirror map in $d="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.13268","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2105.13268/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}