We introduce the notion of hyperbolic orientation of a motivic ring spectrum, which generalises the various existing notions of orientation (by the groups $GL$, ${SL}^c$, $SL$, $Sp$). We show that hyperbolic orientations of $\eta$-periodic ring spectra correspond to theories of Pontryagin classes, much in the same way that $GL$-orientations of arbitrary ring spectra correspond to theories of Chern classes. We prove that $\eta$-periodic hyperbolically oriented cohomology theories do not admit further characteristic classes for vector bundles, by computing the cohomology of the étale classifying space ${BGL}_n$. Finally, we construct the universal hyperbolically oriented $\eta$-periodic commutative motivic ring spectrum, an analogue of Voevodsky's cobordism spectrum $MGL$.

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Motivic Pontryagin 类和双曲方向

我们引入了动机环谱的双曲取向的概念，它概括了各种现有的取向概念（通过群$GL$,${SL}^C$,$SL$,$斯普$）。我们证明了双曲方向$\eta$-周期环谱对应于庞特里亚金类理论，与$GL$-任意环谱的方向对应于陈类理论。我们证明$\eta$-通过计算étale分类空间的上同调，周期双曲导向的上同调理论不承认向量丛的进一步特征类${BGL}_n$。最后，我们构造了通用双曲导向$\eta$-周期性交换动机环谱，Voevodsky 的共边谱的类似物$麦格莱$。