A comparison of motivic and classical stable homotopy theories. (22nd August 2013) - Levine, Marc
Abstract : Let k be an algebraically closed field of characteristic zero. Let c :𝒮ℋ→𝒮ℋ( k ) be the functor induced by sending a space to the constant presheaf of spaces onSm / 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 :𝒮ℋ( k )→𝒮ℋ be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum over k, 𝕊 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 𝕊∈𝒮ℋ; 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 𝒮ℋ( 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.