A vanishing line in the BP〈1〉-Adams spectral sequence

We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulæ for the computation of the E3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

Let A be the mod p Steenrod algebra for p an arbitrary odd prime. In 1962, Liulevicius described hi and bk in Ext*A’*(Zp,Zp) having bigrading (1, sui— 1) and (2, 2pk+1x(p— 1)), respectively. In this paper we prove that for p ≥ 7, n ≥ 4 and \(3 \leqslant s < p - 1, h_0 b_{n - 1} \tilde \gamma _s \in Ext_A^{s + 3,p^n q + sp^2 q + (s - 1)pq + (s - 1)q + s - 3} (Z_p ,Z_p )\) survives to E∞ in the Adams spectral sequence, where q = 2(p — 1).

In (7), we investigated some elements in the Adams spectral sequence for the stable homotopy groups of spheres, and proved that they were never boundaries, for any differential. This paper extends and generalizes these results: we consider more elements than in (7), and also prove that many of them do not survive to the E∞ term, so that they fail to be cycles for some dr, and this differential is therefore non-zero.

For p &gt; 2, is the first positive even-dimensional element in the stable homotopy groups of spheres. A classical theorem of Nishida[1] states that all elements of positive dimension in the stable homotopy groups of spheres are nilpotent. In fact, Toda [4] proved . For p = 3 he showed that while . In [2] the second author computed the first thousand stems of the stable homotopy groups of spheres at the prime 5. One of the consequences of this computation is that while .

This paper proves the existence of a new nontrivial family of filtration three in the stable homotopy group of spheres p2ð p nþpÞð p� 1Þ� 3S which is of order p and is represented by (b0hnþ h1bn� 1Þ in the E 3; 2ð p n þpÞð p� 1Þ 2 -term of the Adams spectral sequence, where pb 5 is a prime and nb 3.

The Adams spectral sequence has been an important tool in research on the stable homotopy of the spheres.In this note we outline new information about a variant of the Adams sequence which was introduced by Novikov [7].We develop simplified techniques of computation which allow us to discover vanishing lines and periodicity near the edge of the E 2 -term, interesting elements in E^'*, and a counterexample to one of Novikov's conjectures.In this way we obtain independently the values of many low-dimensional stems up to group extension.The new methods stem from a deeper understanding of the Brown-Peterson cohomology theory, due largely to Quillen [8]; see also [4].Details will appear elsewhere; or see [ll].When p is odd, the p-primary part of the Novikov sequence behaves nicely in comparison with the ordinary Adams sequence.Computing the £ 2 -term seems to be as easy, and the Novikov sequence has many fewer nonzero differentials (in stems ^45, at least, if p = 3), and periodicity near the edge.The case p = 2 is sharply different.Computing E 2 is more difficult.There are also hordes of nonzero differentials dz, but they form a regular pattern, and no nonzero differentials outside the pattern have been found.Thus the diagram of £4 ( =£oo in dimensions ^17) suggests a vanishing line for E w much lower than that of £2 of the classical Adams spectral sequence [3].It is a pleasure to thank Arunas Liulevicius, my thesis adviser, for his help.In particular, parts of the proofs of Proposition 1 and Theorem 7 are due to him.I am also grateful to many others for their suggestions, and especially to Frank Adams.1.The spectral sequence.The construction of the classical Adams spectral sequence for the spheres [l ] works equally well if the spec-

Ravenel introduced spectra $T(m)$ for $m\ge 0$ interpolating the Brown-Peterson spectrum $BP$ and the sphere spectrum $S$ in Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, Providence, 2004. Since the homotopy groups of $BP$ are well known, it is interesting to study differences among the homotopy groups of $T(m)$'s to study the homotopy groups of spheres. He also introduced the localization functor $L_n$ on the stable homotopy category in "Localization with respect to certain periodic homology theories," Amer. J. Math. 106 (1984), 351–414. To study the difference of $L_nT(m)$'s for a fixed integer $n$, we consider the corresponding chromatic $E_1$-term $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for each $m$, and determine it for $m+1\ge (n-2)(n-1)$ in this paper. The results show that the structures depend on a integer $\Lt[\dfrac{m+1}{n-1}\Rt]$. Here $[x]$ denotes the greatest integer that does not exceed $x$.

A localisation of the category of n-manifolds is introduced by formally inverting the connected sum construction with a chosen n-manifold Y .On the level of automorphism groups, this leads to the stable diffeomorphism groups of n-manifolds.In dimensions 0 and 2, this is connected to the stable homotopy groups of spheres and the stable mapping class groups of Riemann surfaces.In dimension 4 there are many essentially different candidates for the n-manifold Y to choose from.It is shown that the Bauer-Furuta invariants provide invariants in the case Y = CP 2 , which is related to the birational classification of complex surfaces.This will be the case for other Y only after localisation of the target category.In this context, it is shown that the K3-stable Bauer-Furuta invariants determine the S 2 ×S 2 -stable invariants.

The goal of this work is the construction of the analogue to the Adams spectral sequence in cobordism theory, calculation of the ring of cohomology operations in this theory, and also a number of applications: to the problem of computing homotopy groups and the classical Adams spectral sequence, fixed points of transformations of period p, and others.

Algebraic topology of manifolds defined by simple polytopes V M Bukhshtaber and T E Panov Torus actions, combinatorial topology, and homological algebra V M Buchstaber and T E Panov Hirzebruch genera of manifolds with torus action T E Panov The index of an equivariant vector field and addition theorems for Pontryagin classes V M Bukhshtaber and K E Feldman Embedding of compacta, stable homotopy groups of spheres, and singularity theory P M Akhmet'ev Complex cobordism and formal groups Viktor M Buchstaber Massey Products and the Bukhshtaber spectral sequence I V Artel'nykh Toric manifolds and complex cobordisms

We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of $\pi_{51}$ is $\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors (\cite{Isa1}, \cite{IX}, \cite{WX1}) through the 61-stem. The proof of this result uses the $RP^\infty$ technique, which was introduced by the authors in \cite{WX1} to prove $\pi_{61}=0$. This paper advertises this method through examples that have simpler proofs than in \cite{WX1}.

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}.

In this paper, we introduce a four-filtrated version of the May spectral sequence (MSS), from which we study the general properties of the spectral sequence and give a collapse theorem. We also give an efficient method to detect generators of May E1-term E1s,t,b,* for a given (s, t, b, *). As an application, we give a method to prove the non-triviality of some compositions of the known homotopy elements in the classical Adams spectral sequence (ASS).

An application of complex bordism to the stable homotopy groups of spheres

Generalized higher order cohomology operations and stable homotopy groups of spheres