A comparison of motivic and classical homotopy theories

Abstract

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 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. 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.

4th version adds some remarks on combinatorial model categories,, makes some corrections to the appendix on symmetric products, corrects typos updates references

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• Algebraic Kasparov K-theory. II
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• Real motivic and $C_2$-equivariant Mahowald invariants
• The motivic Adams-Novikov spectral sequence at odd primes over $\mathbb{C}$ and $\mathbb{R}$
• Noncommutative stable homotopy and stable infinity categories
• Topological models for stable motivic invariants of regular number rings
• Hermitian $K$-theory, Dedekind $\zeta$-functions, and quadratic forms over rings of integers in number fields
• The geometric Satake equivalence for integral motives
• Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra
• Stable motivic pi_1 of low-dimensional fields
• Correspondences and stable homotopy theory
• Galois equivariance and stable motivic homotopy theory
• Operations on Milnor-Witt K-theory