We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a new computational method that yields a streamlined computation of the first 61 stable homotopy groups, and gives new information about the stable homotopy groups in dimensions 62 through 90. 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.
In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^3$-actions on rational spheres and complex projective spaces, and provide some properties of $S^1$-actions on a general rational complex.
William Browder in his paper "Surgery and the theory of differentiable transformation groups" developed surgery techniques to study semi-free actions of S1 on homotopy spheres, under the additional assumption that the fixed point set is a homotopy sphere. He used this surgery to show how to construct such actions. In this paper, I discussed a similar theory for semi-free actions of S3 on homotopy spheres. An open problem is raised at the end of the paper.
We calculate the rational homotopy and the K(1)-local homotopy of the K(2)-local sphere at the prime 3 and level 2. We use this to verify the chromatic splitting conjecture in this case.
The reader of doi:10.1016/j.topol.2010.08.006 might conjecture that Δ(\barΠ_n)/G is homotopy equivalent to a wedge of spheres for any n>=3 and any subgroup G<S_n. We disprove this by showing that Δ(\barΠ_p)/C_p is not homotopy equivalent to a wedge of spheres for any prime number p>=5.
These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007.
The theory of secondary chomology operations leads to a conjecture concerning the algebra of higher cohomology operations in general. This conjecture is discussed here in detail and its connection with homotopy groups of spheres and the Adams spectral sequence is described.
Group structures of the 2-primary components of the 31-stem homotopy groups of spheres were studied by Oda in 1979. There are, however, two incompletely determined groups. In this paper, our investigation with Toda's composition method gives structures of them.
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 $Cτ$ 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.
Collection of (equivariant) $\rm{PL}$-mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.
We describe the first stem of the stable homotopy groups of spheres by using some Puppe sequences, Thom complexes, K-Theory and Adams operations following the ideas of J. Frank Adams. We also touch upon the second and the third stems in this perspective. Neither spectral squences nor Steenrod operations are used.
We present a new technique for analyzing the v_0-Bockstein spectral sequence studied by Shimomura and Yabe. Employing this technique, we derive a conceptually simpler presentation of the homotopy groups of the E(2)-local sphere for p > 3. We identify and correct some errors in the original Shimomura-Yabe calculation. We deduce the related K(2)-local homotopy groups, and discuss their manifestation of Gross-Hopkins duality.
This paper is an English translation of chapter nine of the book "Adams spectral sequence and stable homotopy groups of spheres" by Jinkun Lin (in Chinese)(Sciences Press, Beijing 2007). In this paper, a sequence of new indecomposable families in the stable homotopy groups of spheres such as h_0h_n,h_nb_n,h_0h_nh_m,h_0(h_mb_{n-1}-b_{m-1}h_n) families was detected. The proof of all the detections was stated in this paper in based on several published papers by the author.
A geometric approach to the stable homotopy groups of spheres is developed in this paper, based on the Pontryagin-Thom construction. The task of this approach is to obtain an alternative proof of the Hill-Hopkins-Ravenel theorem [H-H-R] on Kervaire invariants in all dimensions, except, possibly, a finite number of dimensions. In the framework of this approach, the Adams theorem on the Hopf invariant is studied, for all dimensions with the exception of 15, 31, 63, 127. The new approach is based on the methods of geometric topology.
We bound the volume of the homotopy groups of the 2-local Goodwillie approximations of a sphere in terms of the amount of $2$-torsion in the stable stems, providing a Goodwillie-theoretic refinement of a result of Burklund and Senger (arXiv:2203.00670). At the $2^k$-excisive approximation, this bound is obtained by `multiplying the stable answer by a polynomial of degree $k$'. The main tool is Behrens' Goodwillie-EHP Long Exact Sequence.
Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $\aut_G(S(V)^*)$ denote the space of $G$-equivariant self homotopy equivalences of $S(V)^{*n}$. We show that for any $k \geq 1$ there exists $M>0$ which depends only on $V$ such that $|π_k \aut_G(S(V)^{*n})| \leq M$ is for all $n \gg 0$.
We compute the homotopy groups of the η-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that the η-periodic slice spectral sequence over Q determines the η-periodic slice spectral sequence over all extensions of Q. This leads to a speculation on the role of a "connective Witt-theoretic J-spectrum" in η-periodic motivic homotopy theory.
For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick further developed this for p>=5 by constructing a homotopy fibration S^2n-1 --> T^2n+1(p^r) --> Loop S^2n+1 whose connecting map is degree p^r on the bottom cell. A much simpler construction of such a fibration for p>=3 was given by Gray and the author using new methods. In this paper the new methods are used to start over, first constructing Anick's fibration fo…
In this paper we constructs a new nontrivial family in the stable homotopy groups of spheres $π_{p^nq+2pq+q-3}S$ which is of order $p$ and is represented by $k_0h_{n} \in Ext_A^{3,p^nq+2pq+q}(\mathbb{Z}_p,\mathbb{Z}_p)$ in the Adams spectral sequence, where $p\geq 5$ is an odd prime, $n\geq 3$ and $q=2(p-1)$. In the course of the proof, a new family of homotopy elements in $π_{\ast}V(1)$ which is represented by $β_{\ast}{i^{\prime}}_{\ast}i_{\ast}({h}_n)\in Ext_A^{2,p^nq+(p+1)q+1}(H^{\ast}V(1),\mathbb{Z}_p)$ in the Adams sequence is detected.
In this paper, we study the cohomology of the Morava stabilizer algebra $S(3)$. As an application, we show that for $p \geq 7$, if $s\not \equiv 0, \pm 1 \,\, mod \,p $, $n\not \equiv 1 \,\, mod\, 3$, $n>1$, then $ζ_nγ_s$ is a nontrivial product in $π_*(S)$ by Adams-Novikov spectral sequence, where $ζ_n$ is created by R. Cohen \cite{Co}, $γ_s$ is a third periodic homotopy elements.
The main purpose of this note is to give a proof of the fact that the Toda brackets $<\barν,σ,\barν>$ and $<ν,η, \barσ>$ are not trivial. This is an affirmative answer of the second author's Conjecture (Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs 13(2008), 355-383). The second purpose is to show the relation $\barν_7ω_{15}=ν_7σ_{10}κ_{17}$ in $π^7_{31}$.
In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated c…
In this paper we study $\mathcal M(X)$, the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of $X$, via a map $Ψ$ from $\mathcal M(X)$ into the quotient of $K(X)=[X,BSO]$ by the action of the group of homotopy classes of simple self equivalences of $X$. The map $Ψ$ describes which bundles over $X$ can occur as normal bundles of manifolds in $\mathcal M(X)$. We determine the image of $Ψ$ when $X$ belongs to a certain class of homology spheres. In particular, we find conditions on elements of $K(X)$ that guarantee they are pullbacks of normal bundles of manifolds …
In the paper of Montgomery, D. and Yang, C.T. [5], they discuss the de-suspension of smooth free actions of S1 on (2n+1)-dimensional homotopy spheres. In this paper we discuss the de-suspension of smooth free actions of S3 on (4n + 3)-dimensional homotopy spheres.
We show that an infinite sequence of homotopy 4-spheres constructed by Cappell-Shaneson are all diffeomorphic to S^4. This generalizes previous results of Akbulut-Kirby and Gompf.
We give a combinatorial description of general homotopy groups of $k$-dimensional spheres with $k\geq3$ as well as those of Moore spaces. For $n>k\geq 3,$ we construct a finitely generated group defined by explicit generators and relations, whose center is exactly $π_n(S^k)$.
A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on $\Z/2^{[3]}$--framed immersions, and quaternionic-cyclic structure on $\Z/2^{[4]}$--framed immersions. Using these concepts, we prove that for sufficiently large $n$, $n=2^{\ell}-2$, an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant. Additionally, for $\ell \ge 12$ (i.e., for $n \ge 4094$) an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant if this skew-frame…
Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}^n U_n$ and $S(V_n)$ denote the linear sphere of unit vectors. Then for any $i \geq 0$ the sequence of group $\dots \rightarrow π_i \operatorname{map}^G(S(V_n),S(V_n)) \rightarrow π_i \operatorname{map}^G(S(V_{n+1}),S(V_{n+1})) \rightarrow \dots$ stabilizes with the stable group $\oplus_H ω_i(BW_GH)$ where $H$ runs through representatives of the conjugacy classes of all the isotropy group of…
In this paper we give a formula for the homotopy groups of $(n-1)$-connected $2n$-manifolds as a direct sum of homotopy groups of spheres in the case the $n^{th}$ Betti number is larger than $1$. We demonstrate that when the $n^{th}$ Betti number is $1$ the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore.
We prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres.
We examine the relation between the gauge groups of $\mathrm{SU}(n)$- and $\mathrm{PU}(n)$-bundles over $S^{2i}$, with $2\leq i\leq n$, particularly when $n$ is a prime. As special cases, for $\mathrm{PU}(5)$-bundles over $S^4$, we show that there is a rational or $p$-local equivalence $\mathcal{G}_{2,k}\simeq_{(p)}\mathcal{G}_{2,l}$ for any prime $p$ if, and only if, $(120,k)=(120,l)$, while for $\mathrm{PU}(3)$-bundles over $S^6$ there is an integral equivalence $\mathcal{G}_{3,k}\simeq\mathcal{G}_{3,l}$ if, and only if, $(120,k)=(120,l)$.
We show the "non-existence" results are essential for all the previous known applications of the Bauer-Furuta stable homotopy Seiberg-Witten invariants. As an example, we present a unified proof of the adjunction inequalities. We also show that the nilpotency phenomenon explains why the Bauer-Furuta stable homotopy Seiberg-Witten invariants are not enough to prove 11/8-conjecture.
In this article we show that the $p$-torsion exponent of the stable stems grows sublinearly in $n$ and the $p$-rank of the $E_2$-page of the Adams spectral sequence grows as $\exp(Θ( \log(n)^3))$. Together these bounds provide the first subexponential bound on the size of the stable stems. Conversely, we prove that a certain, precise, version of the failure of the telescope conjecture would imply that the upper bound provided by the Adams $E_2$-page is essentially sharp -- answering the titular question: As big as the fate of the telescope conjecture demands. In an appendix joint with Andrew S…
The purpose of this note is to give a description of the stable homotopy group of some $2$-stage Postnikov systems for a range of dimensions. The considered $2$-stage Postnikov systems has the (weak) homotopy type of a generalized Eilenberg-Mac lane space, then their stable homotopy groups can be calculated by the stable homotopy groups of generalized Eilenberg-Mac lane spaces.
In this note, we compute the image of the $α$-family in the homotopy of the $K(2)$-local sphere at the prime $p=2$ by locating its image in the algebraic duality spectral sequence. This is a stepping stone for the computation of the homotopy groups of the $K(2)$-local sphere at the prime $2$ using the duality spectral sequences.
We survey various approaches to axiomatic stable homotopy theory, with examples including derived categories, categories of (possibly equivariant or localized) spectra, and stable categories of modular representations of finite groups. We focus mainly on representability theorems, localisation, Bousfield classes, and nilpotence.
We give a combinatorial description of homotopy groups of $ΣK(π,1)$. In particular, all of the homotopy groups of the $3$-sphere are combinatorially given.
We compute the 2-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of motivic cohomology and hermitian K-groups.
Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of twisted sector, along with a classical theorem of tom Dieck, allows for a natural definition of stable orbifold homotopy groups, and motivates defining extended unstable orbifold homotopy groups generalizing previous definitions.
For any finite group G, we show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all "higher order structure" of the 2-local G-equivariant stable homotopy category, such as the equivariant homotopy types of function G-spaces. The theorem can be seen as an equivariant version of Schwede's rigidity theorem at the prime 2.
This paper contains an overview of background from stable homotopy theory used by Freed--Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence. Many examples are worked out in detail to illustrate the techniques.
We investigate a new case of rigidity in stable homotopy theory which is the rigidity of the $K(1)$-local stable homotopy category $\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ at $p=2$. In other words, we show that recovering higher homotopy information by just looking at the triangulated structure of $\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ is possible, which is a property that only few interesting stable model categories are known to possess.
We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres…
Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $Σ(2n)$ which has the homotopy type of the $2n$- sphere $\mathbb{S}^{2n}$. Then, this action induces an action of the group $G$ on the top cohomology of $Σ(2n)$. For the family of virtually cyclic groups, we classify all groups which act on $Σ(2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $\mbox{dim}\,Σ(2n)\leq 2n+1$, we study the groups with the virtual cohomological dimension $\mbox{vcd}\,G<\infty$ …
In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over $\mathbb{C}$. In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.
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 [29]), 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 …
