2025/10/11 by Gant, Sebastian, Williams, Ben
#13C10 (Secondary) #14F42 (Primary) #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.10033
Working over an algebraically closed field of characteristic 0, we show that the motivic stable homotopy groups of the sphere spectrum can be determined entirely from the motivic homotopy groups of the p-completed sphere spectra and the motivic cohomology of the ground field, except possibly for the 0 and -1-stems. Using this, we show that the complex realization map from the motivic homotopy group to the classical stable homotopy group is an isomorphism in a range of bidegrees. We apply this to deduce that complex realization also induces isomorphisms on unstable homotopy groups for Stiefel varieties Vr(\mathbbAnk) in a range of bidegrees. This allows a complete solution of the question of when the projection map Vr(\mathbbAnk) → V1(\mathbbAnk) admits a right inverse. Equivalently, this settles the question of when the universal stably-free module of type (n,n-1) admits a free summand of given rank.