Over characteristic 0 fields, the unit map from the motivic sphere spectrum to the special linear cobordism spectrum MSL is an isomorphism on homotopy modules, proven by comparing framed and SL-oriented framed correspondences.
The 0-th stable A^1-homotopy sheaf and quadratic zero cycles
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abstract
We study the 0-th stable A^1-homotopy sheaf of a smooth proper variety over a field k assumed to be infinite, perfect and to have characteristic unequal to 2. We provide an explicit description of this sheaf in terms of the theory of (twisted) Chow-Witt groups as defined by Barge-Morel and developed by Fasel. We study the notion of rational point up to stable A^1-homotopy, defined in terms of the stable A^1-homotopy sheaf of groups mentioned above. We show that, for a smooth proper k-variety X, existence of a rational point up to stable A^1-homotopy is equivalent to existence of a 0-cycle of degree 1.
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The unit map of the algebraic special linear cobordism spectrum
Over characteristic 0 fields, the unit map from the motivic sphere spectrum to the special linear cobordism spectrum MSL is an isomorphism on homotopy modules, proven by comparing framed and SL-oriented framed correspondences.