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 E₂ agrees with the E₂ terms in the Adams-Novikov spectral sequence.

• 1022.bib (251 bytes)
• MotVClass.pdf (365452 bytes) [January 1, 2012]

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