### A comparison of motivic and classical stable homotopy theories, by Marc Levine

Let k be an algebraically closed field of characteristic zero. Let SH(k) be the
motivic stable homotopy category of T-spectra over k, SH the classical stable
homotopy category and 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
π_{n}(E) → π_{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. Let S_{k} be the motivic sphere
spectrum. We show that the Tate-Postnikov tower for S_{k} has Betti
realization which is strongly convergent. This gives a spectral sequence of
algebro-geometric 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.

Marc Levine <marc.levine@uni-due.de>