We show that a triangulated motivic category admits categorical Thom
isomorphisms for vector bundles with an additional structure if and
only if the generalized motivic cohomology theory represented by the
tensor unit object admits Thom classes. We also show that the stable
–derived
category does not admit Thom isomorphisms for oriented vector bundles and, more
generally, for symplectic bundles. In order to do so we compute the first homology
sheaves of the motivic sphere spectrum and show that the class in the coefficient ring of
–homology
corresponding to the second motivic Hopf map
is nonzero,
which provides an obstruction to the existence of a reasonable theory of Thom classes in
–cohomology.
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