On the Zero Slice of the Sphere Spectrum

We prove the motivic analogue of the statement saying that the zero stable homotopy group of spheres is $\mathbf Z$. In topology, this is equivalent to the fact that the fiber of the obvious map from the sphere $S^n$ to the Eilenberg–MacLane space $K(\mathbf Z,n)$ is $(n+1)$-connected. We prove our motivic analogue by an explicit geometric investigation of a similar map in the motivic world. Since we use the model of the motivic Eilenberg–MacLane spaces based on the symmetric powers, our proof works only in zero characteristic.