A comparison of motivic and classical homotopy theories

Let k be an algebraically closed field of characteristic zero. Let SH(k) denote the motivic stable homotopy category of T-spectra over k and SH the classical stable homotopy category. Let c:SH -> SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism {\pi}_n(E)-> {\pi}_{n,0}c(E) for all spectra E. Fix an embedding of k into the complex numbers and let Re:SH(k) -> SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum has Betti realization which is strongly convergent. This gives a spectral sequence "of motivic origin" converging to the homotopy groups of the classical sphere spectrum; this spectral sequence at E_2 agrees with the E_2 terms in the Adams-Novikov spectral sequence.