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Let  k  be an algebraically closed field of characteristic zero. Let  c :ℋ→ℋ( 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 consequence,  c  induces an isomorphismc  *   :   π  n    (  E  )   →   Π   n  ,  0(   c   (  E  )    )    (  k  )for all spectra  E  and all  n ∈ℤ.  Fix an embedding σ: k →ℂ and let  Re   B  :ℋ( k )→ℋ be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum over  k ,   k  , has Betti realization which is strongly convergent. This gives a spectral sequence ‘of motivic origin’ converging to the homotopy groups of the sphere spectrum ∈ℋ; this spectral sequence at  E  2  agrees with the  E  2  terms in the Adams–Novikov spectral sequence after a reindexing. Finally, we show that, for ε a torsion object in ℋ( k ) eff , the Betti realization induces an isomorphism Π  n , 0 (ε)( k )→π  n  ( Re   B  ε) for all  n , generalizing the Suslin–Voevodsky theorem comparing mod  N  Suslin homology and mod  N  singular homology.

A comparison of motivic and classical stable homotopy theories