Motivic Pontryagin classes and hyperbolic orientations

Abstract

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