Complex motivic $kq$-resolutions

Dominic Leon Culver, J. D. Quigley

Published 2019-05-28Version 1

We analyze the $kq$-based motivic Adams spectral sequence over the complex numbers, where $kq$ is the very effective cover of Hermitian K-theory defined over $\mathbb{C}$ by Isaksen-Shkembi and over general base fields by Ananyevskiy-R{\"o}ndigs-{\O}stv{\ae}r. We calculate the ring of cooperations of $kq$ modulo $v_1$-torsion, completely calculate the $0$- and $1$-lines of the $kq$-resolutions, completely determine the $v_1$-periodic complex motivic stable stems, and recover Andrews and Miller's computation of the $\eta$-periodic complex motivic stable stems. As an application, we propose a motivic Telescope Conjecture and outline a program for proving the conjecture in two cases using our calculations. We also propose a model for the complex motivic stable orthogonal J-homomorphism and conjecture its location in the $kq$-resolution.