The geometric diagonal of the special linear algebraic cobordism

[2409.16962] The geometric diagonal of the special linear algebraic cobordism Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Algebraic Topology arXiv:2409.16962 (math) [Submitted on 25 Sep 2024 ( v1 ), last revised 2 Jun 2026 (this version, v4)] Title: The geometric diagonal of the special linear algebraic cobordism Authors: Egor Zolotarev View a PDF of the paper titled The geometric diagonal of the special linear algebraic cobordism, by Egor Zolotarev View PDF HTML (experimental) Abstract: 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 subgroup. The complete answer is given in terms of the special unitary cobordism ring. An important component of the computation is the construction of Pontryagin characteristic numbers with values in the hermitian K-theory. We also construct Chern numbers in this setting, prove the motivic version of the Anderson-Brown-Peterson theorem and briefly discuss classes of Calabi-Yau varieties in the $\mathrm{SL}$-cobordism ring. Comments: 41 pages, final version, fixed a minor sign mistake Subjects: Algebraic Topology (math.AT) ; Algebraic Geometry (math.AG); K-Theory and Homology (math.KT) MSC classes: 14F42, 19G38, 57R77 Cite as: arXiv:2409.16962 [math.AT] (or arXiv:2409.16962v4 [math.AT] for this version) https://doi.org/10.48550/arXiv.2409.16962 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Adv. Math. 493 (2026), Paper No. 110922, 60 pp Related DOI : https://doi.org/10.1016/j.aim.2026.110922 Focus to learn more DOI(s) linking to related resources Submission history From: Egor Zolotarev [ view email ] [v1] Wed, 25 Sep 2024 14:20:03 UTC (40 KB) [v2] Wed, 4 Dec 2024 11:17:14 UTC (40 KB) [v3] Tue, 24 Mar 2026 14:58:49 UTC (51 KB) [v4] Tue, 2 Jun 2026 17:49:29 UTC (51 KB) Full-text links: Access Paper: View a PDF of the paper titled The geometric diagonal of the special linear algebraic cobordism, by Egor Zolotarev View PDF HTML (experimental) TeX Source view license Current browse context: math.AT < prev | next > new | recent | 2024-09 Change to browse by: math math.AG math.KT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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