{"paper":{"title":"Splitting the Madsen-Tillmann Spectra $MT\\theta_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Andrew Senger, Jonathan Sejr Pedersen","submitted_at":"2025-03-13T16:09:17Z","abstract_excerpt":"We prove that the Madsen-Tillmann spectrum $MT\\theta_n$ splits into the sum of spectra $\\Sigma^{-2n}MO\\langle n+1 \\rangle \\oplus \\Sigma^{\\infty-2n}\\mathbb{R} P^\\infty_{2n}$ after Postnikov trunctation $\\tau_{\\leq \\ell}$ for $\\ell = \\lfloor \\frac{n}{2} \\rfloor - 6$. To accomplish this, we prove that the connecting map in a certain fiber sequence is nullhomotopic in this range by an Adams filtration argument. As an application, we compute $H_2(B\\operatorname{Diff}(W^{2n}_{g},D^{2n});\\mathbb{Z})$ up to extensions for $n \\geq 16$ and $g \\geq 7$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.10507","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/2503.10507/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"}