Cooperations in motivic homotopy theory

Abstract

A central task in motivic homotopy theory is to compute the stable motivic homotopy groups of spheres. These groups are organized into periodic layers, each of which being more tractable than the whole. A useful tool for computing the $v_1$-periodic layer is the $\mathrm{kq}$-resolution, a spectral sequence which encodes descent from hermitian K-theory. This spectral sequence takes the form\[ \mathrm{E}^{s,f,w}_1 = \pi^F_{s+f,w}(\mathrm{kq}\otimes \overline{\mathrm{kq}}^{\otimes n}) \implies \pi^F_{**}(\mathbb{S}). \] The primary goal of this thesis is to understand the kq-resolution. To this end, we compute the ring of cooperations for hermitian K-theory $\pi_{**}^F(\mathrm{kq} \otimes \mathrm{kq})$ over the real numbers and all finite fields of characteristic different from 2. As consequences, we determine (up to $v_1$-torsion) the $\mathrm{E}_1$-page of the $\mathrm{kq}$-resolution over these base fields and deduce a spectrum-level splitting for the symplectic K-theory spectrum $\mathrm{ksp} over any base field of characteristic different from 2.