On the Relation of Symplectic Algebraic Cobordism to Hermitian K-Theory
- 作者: Panin I.1, Walter C.2
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隶属关系:
- St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
- Laboratoire J.-A. Dieudonné (UMR 7351 du CNRS), Département de mathématiques
- 期: 卷 307, 编号 1 (2019)
- 页面: 162-173
- 栏目: Article
- URL: https://journals.rcsi.science/0081-5438/article/view/175944
- DOI: https://doi.org/10.1134/S0081543819060099
- ID: 175944
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详细
We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism ϕ: MSp → BO of commutative ring T-spectra which sends the Thom class thMSp to the Thom class thBO. Using ϕ we construct an isomorphism of bigraded ring cohomology theories on the category \({\mathop{\rm Sm}\nolimits} {\mathcal O}p/S,\bar \varphi :{{\mathop{\rm MSp}\nolimits} ^{*,*}}(X,U){ \otimes _{{\rm{MS}}{{\rm{p}}^{4*,0*}}({\rm{pt}})}}{\rm{B}}{{\rm{O}}^{4*,2*}}({\rm{pt}}) \cong {\rm{B}}{{\rm{O}}^{*,*}}(X,U)\). The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSpp,q = MSp1q−p[q], we have an isomorphism \(\bar \varphi :{{\mathop{\rm MSp}\nolimits} _*}^{[*]}(X,U){ \otimes _{{\rm{MSp}}_0^{[2*]}({\rm{pt}})}}{\rm{KO}}_0^{[2*]}({\rm{pt}}) \cong {\rm{K}}{{\rm{O}}_*}^{[*]}(X,U)\), where the KOi[n](X,U) are Schlichting’s hermitian K-theory groups.
作者简介
I. Panin
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
编辑信件的主要联系方式.
Email: paniniv@gmail.com
俄罗斯联邦, nab. Fontanki 27, St. Petersburg, 191023
C. Walter
Laboratoire J.-A. Dieudonné (UMR 7351 du CNRS), Département de mathématiques
编辑信件的主要联系方式.
Email: walter@math.unice.fr
法国, Parc Valrose, Nice Cedex 02, 06108