Toric manifolds and complex cobordisms

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 $x$.

For p > 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 .

Linkings and Pfaff's equations D Bennequin The topology of four-dimensional manifolds Yu P Solov'ev Singularities of systems of rays Vladimir I Arnol'd On the geometry of hyperbolic attractors of smooth cascades Romen V Plykin Surgery on manifolds with finite fundamental groups A F Kharshiladze Embedding of compacta, stable homotopy groups of spheres, and singularity theory P M Akhmet'ev Cell-like maps. The problem of raising dimension A N Dranishnikov and E V Shchepin IMMERSIONS AND EMBEDDINGS OF SMOOTH MANIFOLDS

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.

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.

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

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

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.

Generalized higher order cohomology operations and 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).

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}.

The chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the $E_{2}$ -term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The $E_{1}$ -term $E_{1}^{s,t}(k)$ of the spectral sequence is an Ext group of $BP_{*}BP$ -comodules. There is a sequence of Ext groups $E_{1}^{s,t}(n-s)$ for nonnegative integers $n$ with $E_{1}^{s,t}(0)=E_{1}^{s,t}$ , and there are Bockstein spectral sequences computing a module $E_{1}^{s,*}(n-s)$ from $E_{1}^{s-1,*}(n-s+1)$ . So far, a small number of the $E_{1}$ -terms are determined. Here, we determine the $E_{1}^{1,1}(n-1)=\\operatorname {Ext}^{1}M^{1}_{n-1}$ for $p\\textgreater 2$ and $n\\textgreater 3$ by computing the Bockstein spectral sequence with $E_{1}$ -term $E_{1}^{0,s}(n)$ for $s=1,2$ . As an application, we study the nontriviality of the action of $\\alpha_{1}$ and $\\beta_{1}$ in the homotopy groups of the second Smith-Toda spectrum $V(2)$ .

\S 7. The adjoint of $h_{p}$ and the theorem of Kahn-Priddy.

A proof of a weak form of Serre's theorem on the finiteness of stable homotopy groups of spheres is suggested, which uses neither loop spaces nor spectral sequences.

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.