Isotropic stable motivic homotopy groups of spheres

Abstract

In this paper we explore the isotropic stable motivic homotopy category constructed from the usual stable motivic homotopy category, following the work of Vishik on isotropic motives (see [26]), by killing anisotropic varieties. In particular, we focus on cohomology operations in the isotropic realm and study the structure of the isotropic Steenrod algebra. Then, we construct an isotropic version of the motivic Adams spectral sequence and apply it to find a complete description of the isotropic stable homotopy groups of the sphere spectrum, which happen to be isomorphic to the Ext-groups of the topological Steenrod algebra. At the end, we will see that this isomorphism is not only additive but respects higher products, completely identifying, as rings with higher structure, the cohomology of the classical Steenrod algebra with isotropic stable homotopy groups of spheres.

Cellular objects in isotropic motivic categories

“…the cofiber of τ , are identified with the E 2 -page of the classical Adams-Novikov spectral sequence, while in [8] the motivic spectrum Cτ is provided with an E ∞ -ring structure inducing an isomorphism of rings with higher products between π * * (Cτ ) and the classical Adams-Novikov E 2 -page. A parallel result for isotropic categories was obtained in [25], where the isotropic sphere spectrum X has been equipped with an E ∞ -ring structure inducing an isomorphism of rings with higher products between π * * (X) and the classical Adams E 2 -page. Moreover, in [9] it is described the category of Cτ -cellular spectra, which is proved to be equivalent as a stable ∞-category equipped with a t-structure (see [17]) to the derived category of left BP * BP-comodules concentrated in even degrees, where BP is the Brown-Peterson spectrum and BP * BP its BP-homology.…”

“…In fact, the main result we get in this paper is that, in a similar way to [9], the category of cellular isotropic spectra is equivalent as a stable ∞-category equipped with a t-structure to the derived category of left comodules over the dual of the classical Steenrod algebra A * . So, first, we obtain that the category of isotropic cellular spectra is completely algebraic, which makes it easier to study, and, moreover, it is deeply related to classical topology, as foreseeable from results in [27] and [25].…”

“…In [27], Vishik introduced the isotropic triangulated category of motives and computed the isotropic motivic cohomology of the point, which happens to be strongly related to the Milnor subalgebra. By following this lead, we studied in [25] the isotropic stable motivic homotopy category. In particular, we identified the isotropic motivic homotopy groups of the sphere spectrum with the cohomology of the topological Steenrod algebra, i.e.…”

“…In this work, we intend to follow a similar path for isotropic categories, encouraged by the evident parallelism between the computations of π * * (Cτ ) over complex numbers (see [13] and [8]), on the one hand, and of π * * (X) over flexible fields (see [25]), on the other. More precisely, we have been guided by the idea that studying the isotropic stable motivic homotopy category over a flexible field is similar in some sense to studying the stable ∞-category of Cτ -cellular spectra in the motivic stable homotopy category over complex numbers.…”

“…In Section 2, we provide the main notations that will be followed throughout this work. Then, we move on to Section 3 by recalling isotropic categories and their main properties, mostly referring to results in [27] and [25]. Since we are mainly interested in cellular objects, we recall in Section 4 definitions and some of the main results from [5], which will be useful in the rest of the paper.…”