This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

中文翻译：

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动机欧拉特性和威特值特性类

本文研究了动机设置中的欧拉特征和特征类。我们建立了 Becker-Gottlieb 转移的动机版本，概括了 Hoyois 的构造。计算还原群中最大环面格式的欧拉特征，证明了从约简的广义分裂原理$\运营商名称{GL}_{n}$要么$\运营商名称{SL}_{n}$到最大环面的归一化器（特征为零）。Ananyevskiy 的分裂原理减少了关于向量丛特征类的问题$\运营商名称{SL}$面向，$\unicode[STIX]{x1D702}$- 秩二丛情况的可逆理论。我们改进了环面归一化器分裂原理$\运营商名称{SL}_{2}$帮助计算二阶丛的对称幂的 Witt 上同调中的特征类，然后将其推广以开发具有 Witt 上同调值的特征类的一般演算。