{"paper":{"title":"The geometric diagonal of the special linear algebraic cobordism","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The P1-diagonal of the homotopy groups of special linear algebraic cobordism equals the special unitary cobordism ring after inverting 2 and the exponential characteristic.","cross_cats":["math.AG","math.KT"],"primary_cat":"math.AT","authors_text":"Egor Zolotarev","submitted_at":"2024-09-25T14:20:03Z","abstract_excerpt":"The motivic version of the $c_1$-spherical cobordism spectrum is constructed. A connection of this spectrum with other motivic Thom spectra is established. Using this connection, we compute the $\\mathbb{P}^1$-diagonal of the homotopy groups of the special linear algebraic cobordism $\\pi_{2*,*}(\\mathrm{MSL})$ over a local Dedekind domain $k$ with $1/2\\in k$ after inverting the exponential characteristic of the residue field of $k$. We discuss the action of the motivic Hopf element $\\eta$ on this ring, obtain a description of the localization away from $2$ and compute the $2$-primary torsion sub"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Using this connection, we compute the P1-diagonal of the homotopy groups of the special linear algebraic cobordism π_{2*,*}(MSL) over a local Dedekind domain k with 1/2∈k after inverting the exponential characteristic of the residue field of k. The complete answer is given in terms of the special unitary cobordism ring.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The computation assumes the base ring is a local Dedekind domain containing 1/2 and that the exponential characteristic of the residue field can be inverted without losing the essential structure of the homotopy groups.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Computes the P1-diagonal of π_{2*,*}(MSL) over local Dedekind domains (with 1/2 in k, after inverting exp char) and expresses it in terms of the special unitary cobordism ring, along with related characteristic numbers and a motivic Anderson-Brown-Peterson theorem.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The P1-diagonal of the homotopy groups of special linear algebraic cobordism equals the special unitary cobordism ring after inverting 2 and the exponential characteristic.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"49c9256fcb8325857701f1b27feb1fef85099a80c884a28b3563a061a7d44549"},"source":{"id":"2409.16962","kind":"arxiv","version":4},"verdict":{"id":"6880cd35-3584-46d7-b99c-96361d853ef7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T20:28:05.618567Z","strongest_claim":"Using this connection, we compute the P1-diagonal of the homotopy groups of the special linear algebraic cobordism π_{2*,*}(MSL) over a local Dedekind domain k with 1/2∈k after inverting the exponential characteristic of the residue field of k. The complete answer is given in terms of the special unitary cobordism ring.","one_line_summary":"Computes the P1-diagonal of π_{2*,*}(MSL) over local Dedekind domains (with 1/2 in k, after inverting exp char) and expresses it in terms of the special unitary cobordism ring, along with related characteristic numbers and a motivic Anderson-Brown-Peterson theorem.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The computation assumes the base ring is a local Dedekind domain containing 1/2 and that the exponential characteristic of the residue field can be inverted without losing the essential structure of the homotopy groups.","pith_extraction_headline":"The P1-diagonal of the homotopy groups of special linear algebraic cobordism equals the special unitary cobordism ring after inverting 2 and the exponential characteristic."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.16962/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"}