2019/05/28 by Culver, Dominic Leon, Quigley, J. D.
#14F42 #55P42 #55Q45 #55Q50 #55Q51 #55T15 #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1905.11952
Let kq denote the very effective cover of Hermitian K-theory. We apply the kq-based motivic Adams spectral sequence, or kq-resolution, to computational motivic stable homotopy theory. Over base fields of characteristic not two, we prove that the n-th stable homotopy group of motivic spheres is detected in the first n lines of the kq-resolution, thereby reinterpreting results of Morel and Röndigs-Spitzweck-Østvær in terms of kq and kq-cooperations. Over algebraically closed fields of characteristic 0, we compute the ring of kq-cooperations modulo v1-torsion, establish a vanishing line of slope 1/5 in the E2-page, and completely determine the 0- and 1- lines of the kq-resolution. This gives a full computation of the v1-periodic motivic stable stems and recovers Andrews and Miller's calculation of the η-periodic ℂ-motivic stable stems. We also construct a motivic connective j spectrum and identify its homotopy groups with the v1-periodic motivic stable stems. Finally, we propose motivic analogs of Ravenel's Telescope and Smashing Conjectures and present evidence for both.