Motivic stable homotopy groups via framed correspondences

Abstract

Let k be a perfect field. Using the technique of framed correspondences, we obtain an expression for mapping spaces from suspension spectra to Thom spectra in the motivic stable homotopy ∞-category SH(k). This result allows us to express some stable homotopy groups of Thom spectra in terms of geometric generators and relations, and we apply this approach to study the unit map of the algebraic special linear cobordism spectrum MSL. We introduce SL-oriented framed correspondences and identify non-positive Gm-homotopy groups of MSL with stabilizations of free abelian groups generated by these correspondences, modulo A^1-homotopy. When k has characteristic 0, we show that the unit map 1_k → MSL induces an isomorphism of homotopy modules, by a direct comparison of these abelian groups. As a straightforward corollary, we deduce over a base field of characteristic 0 the known fact that the Chow-Witt groups and the MW-motivic cohomology groups are SL-oriented cohomology theories.