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1. Stable homotopy groups of spheres and motivic homotopy theory

   Isaksen, Daniel C; Wang, Guozhen; Xu, Zhouli (December 2023, European Mathematical Society Press)

   Beliaev, Dmitry; Smirnov, Stanislav (Ed.)

   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. 

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2. Stable homotopy groups of spheres: from dimension 0 to 90

   https://doi.org/10.1007/s10240-023-00139-1  

   Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (June 2023, Publications mathématiques de l'IHÉS)

   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. 

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3. Some smooth circle and cyclic group actions on exotic spheres

   https://doi.org/10.1090/proc/16662  

   Quigley, JD (February 2024, Proceedings of the American Mathematical Society)

   Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to deduce the existence of smooth circle and cyclic group actions on certain exotic spheres. 

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4. Graphing, homotopy groups of spheres, and spaces of long links and knots

   https://doi.org/10.1017/fms.2024.114  

   Koytcheff, Robin (January 2025, Forum of Mathematics, Sigma)

   We study homotopy groups of spaces of long links in Euclidean space of codimension at least three. With multiple components, they admit split injections from homotopy groups of spheres. We show that, up to knotting, these account for all the homotopy groups in a range which depends on the dimensions of the source manifolds and target manifold and which roughly generalizes the triple-point-free range for isotopy classes. Just beyond this range, joining components sends both a parametrized long Borromean rings class and a Hopf fibration to a generator of the first nontrivial homotopy group of the space of long knots. For spaces of equidimensional long links of most source dimensions, we describe generators for the homotopy group in this degree in terms of these Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map which increases source and target dimensions by one. 

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5. Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

   https://doi.org/10.1090/cams/4  

   Hopkins, Michael; Lin, Jianfeng; Shi, XiaoLin Danny; Xu, Zhouli (February 2022, Communications of the American Mathematical Society)

   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. 

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