Adams不同ials在e-族上的应用

不稳定Adams微分的自动化证明

Automated proofs of unstable Adams differentials

本文提出一种计算机辅助方法，利用不稳定莱布尼茨法则和EHP序列自然性计算不稳定Adams谱序列微分，给出球面2-主不稳定同伦群至50-茎的阶数界及证明示例。

类$h_j^3$上的Adams微分

The Adams differentials on the classes $h_j^3$

本文在Adams过滤3中证明了类$h_j^3$（$j \geq 6$）上存在无穷多个非平凡$d_4$-微分，证实Mahowald猜想，并利用$\mathbb{C}$-motivic与$\mathbb{F}_2$-合成同伦理论，同时确定了$h_j^2$和$h_6^2$的存活页。

The $E_3$ page of the Adams spectral sequence

In the early 2000's, Baues computed the secondary Steenrod algebra, the algebra of all secondary cohomology operations. Together with Jibladze, they showed that this gives an algorithm that computes all Adams $d_2$ differentials for the sphere. The goal of this paper is to reinterpret their results in the language of synthetic spectra in order to achieve stronger computational results. Using this, we obtain an algorithm that computes hidden extensions on the $E_3$ page that jump by one filtration, in addition to the $d_2$ differentials of Baues--Jibladze. We then implement and run this algorit…

CW谱之间Adams微分与扩张问题的机器证明

Machine Proofs for Adams Differentials and Extension Problems among CW Spectra

本研究通过计算方法获取大量Adams微分与扩张，构建含49个CW谱等信息的Zenodo数据集并提供机器证明，结合人类特设论证解决了126维最后的凯瓦雷不变量问题。

Motivic Stable Homotopy and the Stable 51 and 52 Stems

We establish a differential $d_2(D_1)=h_0^2h_3g_2$ in the $51$-stem of the Adams spectral sequence at the prime $2$, which gives the first correct calculation of the stable 51 and 52 stems. This differential is remarkable since we know of no way to prove it without recourse to the motivic Adams spectral sequence. It is the last undetermined differential in the range of the first author's detailed calculations of the $n$-stems for $n<60$ [6]. This note advertises the use of the motivic Adams spectral sequence to obtain information about classical stable homotopy groups.

The eta-local motivic sphere

We compute the h_1-localized cohomology of the motivic Steenrod algebra over C. This serves as the input to an Adams spectral sequence that computes the motivic stable homotopy groups of the eta-local motivic sphere. We compute some of the Adams differentials, and we state a conjecture about the remaining differentials.

Computation of the E_3-term of the Adams spectral sequence

An algorithm is described giving effective determination of the second differential in the Adams spectral sequence. The algorithm is based on the notion of secondary derived functor, and on the explicit algebraic model of the groupoid enriched category of stable maps and stable tracks between Eilenberg-Mac Lane spaces.

Detection of some elements in the stable homotopy groups of spheres

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.

Congruences between modular forms given by the divided beta family in homotopy theory

We characterize the 2-line of the p-local Adams-Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p > 3. We give a similar characterization of the 1-line, reinterpreting some earlier work of A. Baker and G. Laures. These results are then used to deduce that, for l a prime which generates the p-adic units, the spectrum Q(l) detects the alpha and beta families in the stable stems.

Burklund-Lin-Wang-Xu 方法在 Tau 余纤维形式体系中的应用及对等变片微分的应用

Burklund-Lin-Wang-Xu Methods in the Cofiber-of-Tau Formalism and Applications to Equivariant Slice Differentials

通过 Burklund-Isaksen-Pstragowski-Wang-Xu 的 tau 余纤维形式体系重新研究谱序列理论，定义了过滤谱间映射的隐藏扩张，将广义 Leibniz 规则和 Mahowald 技巧推广到更广泛设置，并应用于 C4-等变片谱序列得到新的“异种转移”微分。

The triviality of the 61-stem in the stable homotopy groups of spheres

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-…

On the May Spectral Sequence at the prime 2

We make a conjecture about the whole $E_2$ page of the May spectral sequence in terms of generators and relations and we prove it in a subalgebra which covers a large range of dimensions. We show that the $E_2$ page plays a universal role in the study of Massey products in commutative DGAs. We conjecture that the $E_2$ page is nilpotent free and also prove it in this subalgebra. We compute all the $d_2$ differentials of the generators in the conjecture and construct maps of spectral sequences which allow us to explore Adams vanishing line theorem to compute differentials in the May spectral se…

Z/4×Z/4 群的 Gromov-Lawson-Rosenberg 猜想

The Gromov-Lawson-Rosenberg Conjecture for Z/4xZ/4

本研究通过 Adams 谱序列计算分类空间连通实 k-同调，结合两类α不变量核检测定理，证明了Z/4×Z/4群的Gromov-Lawson-Rosenberg猜想，同时确定了相关谱序列微分并研究了对应子Hopf代数的谱序列上积结构。