An explicit KO ‐degree map and applications
An explicit KO ‐degree map and applications
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明确的 KO 度图和应用
DOI：
10.1112/topo.12007
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发表时间：
2014
影响因子：
1.1
通讯作者：
J. Fasel
中科院分区：
文献类型：
--
作者：
A. Asok;J. Fasel
本文的目的是研究不稳定A1‐同伦理论中从动力球谱到厄米K‐理论谱的单位映射的类比，即厄米K‐理论中的度映射。我们证明了从奇维分裂光滑仿射二次曲面到厄米K理论中出现的伯特周期性空间的几何模型的显式映射“Suslin矩阵”在适当的意义上稳定于单元映射。作为应用，我们推导出i≤3时的KiMW(F)=GWii(F)，这可以看作是描述场K2的Matsumoto著名定理的推广。这些结果提供了旨在计算n大于或等于4的圈πnA1（An≠0）的程序中的第一步。
The goal of this note is to study the analog in unstable A1 ‐homotopy theory of the unit map from the motivic sphere spectrum to the Hermitian K‐theory spectrum, that is, the degree map in Hermitian K‐theory. We show that ‘Suslin matrices’, which are explicit maps from odd‐dimensional split smooth affine quadrics to geometric models of the spaces appearing in Bott periodicity in Hermitian K‐theory, stabilize in a suitable sense to the unit map. As applications, we deduce that KiMW(F)=GWii(F) for i⩽3 , which can be thought of as an extension of Matsumoto's celebrated theorem describing K2 of a field. These results provide the first step in a program aimed at computing the sheaf πnA1(An∖0) for n⩾4 .