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Abstract We study the$$\mathbb {F}_2$$ $F_2$-synthetic Adams spectral sequence. We obtain new computational information about$$\mathbb {C}$$ $C$-motivic and classical stable homotopy groups. 

We describe in terms of generators and relations the ring structure of the $RO\left(C_2\right)$-graded $C_2$-equivariant stable stems ${\mathrm{\pi <\#comment/>}}_{\star <\#comment/>}^{C_2}$modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational $C_2$-equivariant sphere ${\mathrm{\pi <\#comment/>}}_{\star <\#comment/>}^{C_2}\left({\mathbb{S}}_{\mathbb{Q}}\right)$. 

We consider the problem of computing the stable homotopy groups of spheres, including applications and history. We describe a new technique that yields streamlined computations through dimension 61 and gives new computations through dimension 90 with very few exceptions. We discuss questions and conjectures for further study, including a new approach to the computation of motivic stable homotopy groups over arbitrary base fields. We provide complete charts for the Adams spectral sequence through dimension 90. 

Abstract Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic Adams spectral sequence for $$C\tau $$ C τ and the algebraic Novikov spectral sequence for $$BP_{*}$$ B P ∗ , we compute the classical and motivic stable homotopy groups of spheres from dimension 0 to 90, except for some carefully enumerated uncertainties. 

We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)c♡ when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)cell,c♡ as the category of even graded MU2∗MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum 1c=0 is algebraic. Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just C; to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov–Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k. 

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ⁡ ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ⁡ ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ⁡ ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence.