A comparison of motivic and classical homotopy theories

TL;DR

This paper demonstrates the full faithfulness of the constant presheaf functor from classical to motivic stable homotopy categories over algebraically closed fields of characteristic zero, and explores the Betti realization of the motivic sphere spectrum.

Contribution

It proves the functor from classical to motivic homotopy categories is fully faithful and analyzes the Betti realization of the motivic sphere spectrum's slice tower.

Findings

The functor c:SH -> SH(k) is fully faithful.

The Betti realization of the slice tower converges strongly.

A spectral sequence of motivic origin converges to classical homotopy groups, matching the E_2 terms of the Adams-Novikov spectral sequence.

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 {\pi}_n(E)-> {\pi}_{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.