Bernhard Hanke
A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r is less than or equal to k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non simply connected spaces.
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Group actions on spheres with rank one isotropy
研究秩为二的有限群G，通过特定群论条件构造具有秩一p子群的G-CW复形X，其固定集为同调球面，提供新的非线性G-CW复形例子。
Group actions on spheres with rank one prime power isotropy
本文研究了秩为二的有限群G是否能构造出同调于球面的有限G-CW复形，其等变秩为一的素数幂，当且仅当G不包含Qd(p)的p'-因子。
It is proved that there exists an integer $L$ such that a framed manifold of dimension $2^l-2$, $l\le L$ has the trivial Kervaire Invariant.
We describe the distribution of infinite groups in the $RO(G)$-graded stable homotopy groups of spheres for a finite group $G$.
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $Π_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (…
We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.
Let p be an odd prime. We prove that every rank three p-group acts freely and smoothly on a product of three spheres. To construct this action, we first prove a generalization of a theorem of L\" uck and Oliver on constructions of G-equivariant vector bundles. We also give some other applications of this generalization.
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.
Collection of (equivariant) $\rm{PL}$-mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.
Non-linear proper Fredholm maps and the stable homotopy groups of spheres
本文利用球面稳定同伦群分类 Hilbert 空间间的非线性真 Fredholm 映射，给出满射及其非平凡核，并证明有向情形与稳定同伦群一一对应。
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.
The stable systolic category of a closed manifold M indicates the complexity in the sense of volume. This is a homotopy invariant, even though it is defined by some relations between homological volumes on M. We show an equality of the stable systolic category and the real cup-length for the product of arbitrary finite dimensional real homology spheres. Also we prove the invariance of the stable systolic category under the rational equivalences for orientable 0-universal manifolds.
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.
In this paper we show that most rank two groups act freely on a finite homotopy product of two spheres. This makes new progress on a conjecture by Benson and Carlson which states that a finite group G acts freely on a finite complex with the homotopy type of n spheres if the rank of G is less than or equal to n. Recalling that Qd(p) is the semidirect product of a rank two elementary abelian p-group with SL(2,p), we show that a rank two finite group acts freely on a finite CW-complex homotopic to the product of two spheres if, it does not contain any subquotient isomorphic to Qd(p) for any odd …
Classical stable homotopy groups of spheres via $\mathbb{F}_2$-synthetic methods
本文利用$\mathbb{F}_2$-合成Adams谱序列，获得了$\mathbb{C}$-动机与经典稳定同伦群的新计算信息。
A generalized Kervaire Problem in stable homotopy groups of spheres
本文给出了球面稳定同伦群中 Mahowald 元素的初等自包含描述，并以此构造证明了作者此前提出的广义 Kervaire 问题得到了肯定的解决。
A geometrical approach toward stable homotopy groups of spheres. A Desuspension Theorem (revisited)
该研究采用几何方法研究球面稳定同伦群，阐述并证明了$Π_{2^l-2}$族的去悬垂定理，证明是对作者2010年预印本中定理的小幅修订。
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.
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.
The zeroth stable homotopy groups of motivic spheres over the integers
本文确定了整数上动机球面谱的第零积分Milnor-Witt茎，并建立了多种动机谱的细胞性及混合特征Dedekind域上的绝对纯性结果。
New infinite families in the stable homotopy groups of spheres
该研究识别出球面稳定同调群中七个新的192周期无限族元素，并揭示了其在局部化后的非平凡性质及2- torsion和2-可除性的新信息。
The rational homotopy groups of virtual spheres for rank 1 compact Lie groups
本文通过计算1阶紧李群的虚拟球的有理同调群，展示了代数模型在G-谱类中的有效性，并揭示了tom Dieck分裂在脱悬时的失效问题。
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.
Let $G$ be a connected Lie group of rank one. In this paper the existence of free actions of group $G$ on spheres, real projective spaces and lens spaces has been studied. Most of the results have been obtained for finitistic spaces with cohomology ring isomorphic to the cohomology ring of these spaces. The cohomology algebra of the orbit space has been determined in the case of free $G$-action. The non-existence of $G$-equivariant maps from spheres to cohomology spheres has also been established.
We prove that the 2-primary $π_{61}$ is zero. As a consequence, the Kervaire invariant element $θ_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, θ_4, θ_4, 2\rangle$. Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the only ones are $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-…
We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.
Loop Space Splittings for Codimensional Sphere Bundles
本文在适当的连通性和示性类假设下，建立了实向量丛的球面丛的两个环路分解定理，分别处理闭光滑n-流形上秩为(n-1)和(n-3)的向量丛的球面丛。
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 …
We establish an isomorphism between the stable homotopy groups of the 2-completed motivic sphere spectrum over the real numbers and the corresponding stable homotopy groups of the 2-completed Z/2-equivariant sphere spectrum, in a certain range of dimensions.
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…
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.
We show that every rank two $p$-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group $G$ on a manifold $M$, we construct a smooth free action on $M \times \bbS ^{n_1} \times \dots \times \bbS ^{n_k}$ when the set of isotropy subgroups of the $G$-action on $M$ can be associated to a fusion system satisfying certain properties. Another consequence of this construction is that if $G$ is an (almost) extra-special $p$-group of rank $r$, then it acts freely and smoothly on a product of $r$ spheres.
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…
We study the singular cohomology of the moduli space of rank 2 parabolic bundles on a Riemann surface where the weights are all 1/4. We give a formula, based on work of Boden, for the Poincaré polynomial of this moduli space in general, and deduce a complete set of relations in the cohomology ring for the case when the genus is zero. These relations are recursive in the number of parabolic points (which must be odd), and are analogous to the relations appearing in the cohomology ring for the non-parabolic story, which are recursive in the genus. Our methods use techniques from a paper of Weits…
The stable cohomology of self-equivalences of connected sums of products of spheres
本文确定了球面乘积连通和的同伦自等价稳定分类空间的上同调，并将其用李图复形同调表达。
We prove that, for p an odd prime, every finite p-group of rank 3 acts freely on a finite complex X homotopy equivalent to a product of three spheres.
Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the (persistent homology) rank functions. For a point pattern $X$ we construct a filtration of spaces by taking the union of balls of radius $a$ centered on points in $X$, $X_a = \cup_{x\in X}B(x,a)$. The rank function $β_k(X):{\{(a,b)\in \mathbb{R}^2: a\leq b\}} \to \mathbb{R}$ is then defined by $β_k(X)(a,b) = rank ( ι_*:H_k(X_a) \to H_k(X_b))$ where $ι_*$ is the induced map…
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…
Equivariant formality of corank-one isotropy actions and products of rational spheres
本文完全刻画了满足秩差为1且左作用等变形式化的连通李群对，并修正与扩展了具有奇偶维球面乘积有理同伦型的齐性商分类。
We construct a lift of the $p$-complete sphere to the universal height $1$ higher semiadditive stable $\infty$-category tsade-$1$ of Carmeli--Schlank--Yanovski, providing a counterexample, at height $1$, to their conjecture that the natural functor from tsade-$n$ to $\mathrm{Sp}_{T(n)}$ is an equivalence. We then record some consequences of the construction, including an observation of T. Schlank that this gives a conceptual proof of a classical theorem of Lee on the stable cohomotopy of Eilenberg--MacLane spaces.
Nakajima introduced a resolution of singularities of the Donaldson-Uhlenbeck completion of the moduli space of based instantons over $S^4$. For $k\leq 4$, we extend this result to $\mathbb P^2$ and compute, in the rank-stable limit, the homotopy type of these spaces by showing that, in this limit, these constructions yield universal bundles.
Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres
本文通过BP合成谱和Adams-Novikov谱序列证明了β₁的5重积非零而6重积为零，并推广至其他144周期家族。
We study the moduli space of rank stable based instantons over a connected sum of q copies of CP^2. For c_2=1 we give the homotopy type of the moduli space. For c_2=2 we compute the cohomology of the moduli space.
We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-$k$ graphs gives rise to pullbacks of the associated $C^*$-algebras. We describe a combinatorial version of the connected-sum operation and apply it to the rank-2-graph realisations of the four basic surfaces to deduce that every compact 2-manifold is the topological realisation of a ran…
Given a vector bundle, its (stable) order is the smallest positive integer n such that the n-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bun- dle over configuration spaces of Euclidean spaces have been studied by F.R. Cohen, R.L. Cohen, N.J. Kuhn and J.L. Neisendorfer [5], F.R. Cohen, M.E. Mahowald and R.J. Milgram [7], and S.W. Yang [17]. Moreover, the order and the stable order of the canonical vec- tor bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied …
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