  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.
