M. J. Hopkins, J. H. Palmieri, J. H. Smith
We show that in a generalized Adams spectral sequence, the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property.
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We establish a motivic version of Adams' vanishing line of slope 1/2 in the cohomology of the motivic Steenrod algebra over the complex numbers.
We show that in a generalized Adams spectral sequence, the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property.
We study the Hurewicz map h from the homotopy groups of a spectrum X to the R-homology of its 0th space X(0), where R is a connective commutative S-algebra. We prove that the decreasing filtration of the domain of h associated to an R-based Adams resolution is compatible with the augmentation ideal filtration of the range associated to the suspension spectrum of X(0)_+, an augmented commutative S-algebra. The proof makes use of the interplay of this filtration with Topological Andre Quillen Homology. An application is a Connectivity Theorem: Localize at a prime p and suppose X is (c-1)-connect…
A $v_1$-banded vanishing line for the mod 2 Moore spectrum
本研究基于Burklund等人的工作，证明模2 Moore谱$C(2)$的Adams谱序列在斜率1/5的直线上方于$E_5$页坍塌并刻画存活类，完成了Mahowald1970年宣布未证结果的证明。
Building on work of Stolz, we prove for integers $0 \le d \le 3$ and $k>232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal-Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal-Wi…
We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case and the calculation of its E_2-term involves the cohomology of certain `brave new Hopf algebroids' E^R_*E. In working out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum. We show that the Adams Spectral Sequence for S_R based on a commu…
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…
We construct a generalized Witten genus for spin$^c$ manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin$^c$ manifolds called string$^c$ manifolds. We also construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalized Witten genus and the mod 2 Witten genus on string$^c$ and string (generalized) complete intersections in (product of) complex projective spaces respectively.
On the generalized brace product: relation to $H$-splitting of loop space fibrations \& the $J$-homomorphism
本文研究了广义brace积与loop空间纤维的H-分裂及J-同态的关系，通过有理同调理论证明了广义brace积的消失条件，并应用该结果分析了球面束等纤维的同伦类型。
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…
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。
We prove that the Morava-$K$-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a $p$-local finite Postnikov system with vanishing $(n+1)$st homotopy group.
For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by proving the existence of truncated versions of resolution or $E_2$-model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah …
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