{"paper":{"title":"The spinorial \\tau-invariant and 0-dimensional surgery","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bernd Ammann (IECN), Emmanuel Humbert (IECN)","submitted_at":"2006-07-27T15:42:42Z","abstract_excerpt":"Let $M$ be a compact manifold with a metric $g$ and with a fixed spin structure $\\chi$. Let $\\lambda\\_1^+(g)$ be the first non-negative eigenvalue of the Dirac operator on $(M,g,\\chi)$. We set $$\\tau(M,\\chi):= \\sup \\inf \\lambda\\_1^+(g)$$ where the infimum runs over all metrics $g$ of volume 1 in a conformal class $[g\\_0]$ on $M$ and where the supremum runs over all conformal classes $[g\\_0]$ on $M$. Let $(M^#,\\chi^#)$ be obtained from $(M,\\chi)$ by 0-dimensional surgery. We prove that $$\\tau(M^#,\\chi^#)\\geq \\tau(M,\\chi).$$ As a corollary we can calculate $\\tau(M,\\chi)$ for any Riemann surface "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607716","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}