Stable Homotopy Groups Of Spheres Research Articles

Stable homotopy groups of spheres: from dimension 0 to 90

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 Ctau and the algebraic Novikov spectral sequence for BP_{*}, we compute the classical and motivic stable homotopy groups of spheres from dimension 0 to 90, except for some carefully enumerated uncertainties.

Ranks of 𝑅𝑂(𝐺)-graded stable homotopy groups of spheres for finite groups 𝐺

We describe the distribution of infinite groups within the R O ( G ) RO(G) -graded stable homotopy groups of spheres for a finite group G G .

A new family in the stable homotopy groups of spheres

In this paper, the nontriviality of homotopy elements β 1 ϱ s \beta _1\varrho _s in the p p -primary component of stable homotopy groups of spheres is shown by virtue of the Adams spectral sequence and the May spectral sequence, where ϱ s \varrho _s is represented by b 0 g 0 γ ~ s b_0g_{0}\widetilde {\gamma }_{s} in the Adams spectral sequence, p p is a prime number greater than 5 5 , and 3 ≤ s > p 3\leq s>p .

On the product [formula omitted] in the stable homotopy groups of spheres

Let p be a prime greater than 5. In the present article, we show that the products α1β12β2γt are non-trivial in the p-primary components of the stable homotopy groups of spheres for 4<t<p.

Tmf–based Mahowald invariants

The $2$-primary homotopy $\beta$-family, defined as the collection of Mahowald invariants of Mahowald invariants of $2^i$, $i \geq 1$, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper, we calculate $\mathit{tmf}$-based approximations to this family. Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence for the Tate construction of $\mathit{tmf}$ with trivial $C_2$-action and Behrens' filtered Mahowald invariant machinery.

Higher associativity of Moore spectra

The Moore spectrum Mp(i) is the cofiber of the pi map on the sphere spectrum. For a fixed p and n, we find a lower bound on i for which Mp(i) is guaranteed to be n-fold associative. This bound depends on the stable homotopy groups of spheres.

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

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence ofPin⁡(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 thePin⁡(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 fromBPin⁡(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 thejj-based Atiyah–Hirzebruch spectral sequence.

Isotropic stable motivic homotopy groups of spheres

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 [28]), 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.

The special fiber of the motivic deformation of the stable homotopy category is algebraic

For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/\tau$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{-}\mathbf{Comod}}^{ev})$ as stable $\infty$-categories equipped with $t$-structures. As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $\widehat{S^{0,0}}/\tau$, which converges to the motivic homotopy groups of $\widehat{S^{0,0}}/\tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams-Novikov $E_2$-page for the sphere spectrum $\widehat{S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the 90-stem, with ongoing computations into even higher dimensions.

An extension in the Adams spectral sequence in dimension 54

We establish a hidden extension in the Adams spectral sequence converging to the stable homotopy groups of spheres at the prime 2 in the 54-stem. This extension is exceptional in that the only proof we know proceeds via Pstragowski's category of synthetic spectra. This was the final unresolved hidden 2-extension in the Adams spectral sequence through dimension 80. We hope this provides a concise demonstration of the computational leverage provided by $\mathbb{F}_2$-synthetic spectra.

Stable homotopy groups of spheres

We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.

О свойствах группы кобордизма стабильно-оснащённых погружений в коразмерности k

Изучается понятие „intermediate bordism group" , которое было введено П. Дж. Экклзом для исследовании фильтраций в стабильных гомотопических группах сфер. Введено новое понятие группы кобордизма стабильно-оснащенных погружений. Строится представляю щее пространство для новых групп и вычисляются ранги этих групп кобордизма. Инварианты Хопфа и гомоморфизм Кана-Придди обобщаются на группы кобордизма стабильно-оснащенных погружений.

Real orientations of Lubin–Tate spectra

We show that Lubin–Tate spectra at the prime 2 are Real oriented and Real Landweber exact. The proof is by application of the Goerss–Hopkins–Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for $$E_n$$ with its $$C_2$$ -action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $$C_2$$ -fixed points.

A note on the products $\alpha_1\beta_2\gamma_t$ and $\beta_1^{r+1}\beta_2\gamma_t$ in the stable homotopy of spheres

In the stable homotopy groups of spheres, we have Greek letter elements due to J. F. Adams [2], L. Smith [12] and H. Toda [13]. Here we study the nontriviality of certain products of the first alpha element, the first and the second beta elements and a gamma element in the homotopy groups.

The motivic Mahowald invariant

The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. We study the Mahowald invariant in the setting of motivic stable homotopy theory over $Spec(\mathbb{C})$. We compute a motivic version of the $C_2$-Tate construction for various motivic spectra, and show that this construction produces "blueshift" in these cases. We use these computations to show that the Mahowald invariant of $\eta^i$, $i \geq 1$, is the first element in Adams filtration $i$ of the $w_1$-periodic families constructed by Andrews ~\cite{And14}. This provides an exotic periodic analog of Mahowald and Ravenel's computation ~\cite{MR93} that the classical Mahowald invariant of $2^i$, $i \geq 1$, is the first element in Adams filtration $i$ of the $v_1$-periodic families constructed by Adams ~\cite{Ada66}.

Hurewicz images of real bordism theory and real Johnson–Wilson theories

We show that the Hopf elements, the Kervaire classes, and the κ¯-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the C2-fixed points of the Real bordism spectrum. A subset of these families is detected by the C2-fixed points of Real Johnson–Wilson theory ER(n), depending on n. In the proof, we establish an isomorphism between the slice spectral sequence and the C2-equivariant May spectral sequence of BPR.

A ϖn-Related Family of Homotopy Elements in the Stable Homotopy of Spheres

To determine the stable homotopy groups of spheres π*(S) is one of the central problems in homotopy theory. Let p be a prime greater than 5. The authors make use of the May spectral sequence and the Adams spectral sequence to prove the existence of a ϖn-related family of homotopy elements, β1ωnγs, in the stable homotopy groups of spheres, where n > 3, 3 ≤ s < p − 2 and the ϖn-element was detected by X. Liu.

Singularities and stable homotopy groups of spheres. I

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the stable homotopy groups of spheres.

Singularities and stable homotopy groups of spheres. II

We establish a connection between Morin singularities and stable homotopy groups of spheres. This connection allows us to describe how the images of singularity strata behave around the image of a more complicated stratum.

A nontrivial homotopy element of order p2 detected by the classical Adams spectral sequence

Let p be an odd prime. The authors detect a nontrivial element a p of order p2 in the stable homotopy groups of spheres by the classical Adams spectral sequence. It is represented by $$a_0^{p - 2} h_1 \in Ext_A^{p - 1,pq + p - 2} (\mathbb{Z}/p,\mathbb{Z}/p)$$ in the E2-term of the ASS and meanwhile p · a p is the first periodic element α p .