Misha Kornev

Misha Kornev

PhD student in mathematics

Stable Homotopy Groups Via Motivic Ones

3 minute read

Published:

Briefly, there is a wonderful application of motivic homotopy theory to Agriculture classic Algebraic Topology. Classical and motivic Adams-Novikov spectral sequences coincide in the case of algebraic closed fields with zero characteristic (Marc Levine). This connection allowed to eliminate misunderstanding and mistakes in classical calculations. I will follow the Marc Levin overview.

Now, recall some basic facts in the classical setting. The Thom construction builds spectra starting with universal vector bundles, since the one “jumps out” a suspension whenever a trivial bundle added. For example, $\mathrm{Th}(V\oplus \mathbf{1}) = \Sigma^2\mathrm{Th}(V)$, where $V$ a complex bundle, $\mathbf{1}$ trivial rank one bundle.

Due to Serre it is known that all stable homotopy groups are finite. In particular, the stable Hurewicz homomorphism $\mathbb{S}\to \mathrm{EM}(\mathbb{Z})$ from the sphere spectrum to the Eilenberg-MacLane spectrum induces an isomorphism on rational homotopy groups. It means that the forgetful functor $\mathsf{D}(\mathsf{Ab})\to \mathsf{S}\mathcal{H}$ from the derived category of chain complexes of abelian groups to the stable category of spectra is an equivalence of triangulated categories after the rationalization. The Adams and Adams-Novikov spectral sequences serve as standard tools for calculations of stable homotopy groups.

Now, consider the motivic setting. Recall, that the original motivation of motivic homotopy theory is based on doing homotopy theory for algebraic varieties, with the affine line being like a segment. The last one is not an algebraic variety — that is the point. The motivic stable category can be constructed in terms of the category of sheaves and their localizations. Let $S$ be a fixed good enough scheme. Consider the category $\mathsf{Spc}(S)$ of functors

\begin{equation*} \mathcal{X}: \mathsf{Sm}/S\to\mathsf{Spc} \end{equation*}

from the over category $\mathsf{Sm}/S$ of schemes over $S$ to the category of spaces (e. g., $\mathsf{Top}$ or $s\mathsf{Set}$). For such functors, one can easily define (co-)limits, a suspension, a smash product just pointwise. To construct sheaves from presheaves, we should choose a Grothendieck topology $\tau$ on the category $\mathsf{Sm}/S$ (e. g., the Nisnevich or étale one). It allows us to define weak equivalences, with the resulting localized category being denoted by $\mathcal{H}^{\tau}$. Now, let us to perform another localization, all projections $X\times \mathbb{A}^1\to X$ to be weak equivalences (so, $\mathbb{A}^1$ becomes a contractible space). We get a category $\mathcal{H}^{\tau}(S)$ of motivic spaces.

In the setting of $\mathcal{H}^{\tau}(S)$, one can define the Thom space for any vector bundle over a variety $X$ (by the Yoneda Lemma, thinking about $X$ as a presentable sheaf) via projectivizations: $\mathrm{Th}(V) := \mathbb{P}(V\oplus\mathbf{1})/\mathbb{P}(V)$. Particularly, if we take a trivial vector bundle over $X$, then we will have a “jumping out” suspension. The last one is not an iterated smash with an ordinary circle, but a motive projective line (i. e., a smash $S^1\wedge\mathbb{G}_m$ between an ordinary circle and an algebraic torus) instead.

Moreover, one can define 2-parametric family of motivic spheres

\begin{equation*} S^{a + b, b} := S^a\wedge\mathbb{G}_m^{\wedge b} \end{equation*}

For instance, $\mathbb{P}^1 = S^{2, 1}$. They give a bigraded homotopy group sheaf $\pi_{a, b}(X)$ for a motivic space $X$.

As in the classical setting, the suspension and loop functors (with respect to the motivic projective line $\mathbb{P}^1$) are adjoint and inverse to each other. So, one can form a motivic sphere spectrum. This spectrum corresponds to some (co-)homology theory.

Now, the point of this post: over an algebraic closed field with 0 characteristic, there is a functor $c: \mathsf{S}\mathcal{H}\to \mathsf{S}\mathcal{H}(k)$ from the classical category of spectra to the motivic category spectra over the algebraic point $\mathrm{Spec}(k)$. In particular, this functor induces an isomorphism

\begin{equation*} \pi_n(E)\cong \pi_n^{\mathbb{A}^1}(E\wedge c(\mathbb{S}))(k) \end{equation*}

Putting E to be $\mathbb{S}$, the sphere spectrum, we get:

\begin{equation*} \pi_n(\mathbb{S}) = \pi_n^{\mathbb{A}^1}(\mathbb{S}_k)(k) \end{equation*}

References

Levine, Marc. (2016). An Overview of Motivic Homotopy Theory. Acta Mathematica Vietnamica. 41. 10.1007/s40306-016-0184-x.

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