{"paper":{"title":"Enumerating stably trivial vector bundles with higher real $K$-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Hood Chatham, Morgan Opie, Yang Hu","submitted_at":"2024-03-07T18:36:21Z","abstract_excerpt":"Given positive integers $r$ and $c$, let $\\phi(r,c)$ denote the number of isomorphism classes of complex rank $r$ topological vector bundles on $\\mathbb{CP}^{r+c}$ that are stably trivial. We compute the $p$-adic valuation of the number $\\phi(r,c)$ for all pairs $r$ and $c$ such that $c \\leq \\operatorname{min}\\{r,2p-3\\}$. We also give some systematic lower bounds for $p$-divisibility of $\\phi(r,c)$ when $c<2p^2-p-2$, and detect some nontrivial $p$-divisibility for larger $c$. As an additional application of our methods, we find new $p$-torsion in unstable homotopy groups of unitary groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.04733","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/2403.04733/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"}