The BP-Coaction for Projective Spaces

1978 ◽  
Vol 30 (01) ◽  
pp. 45-53 ◽  
Author(s):  
Donald M. Davis
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Projective Spaces ◽  
Homology Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
New Information ◽  
Image Position

The Brown-Peterson spectrum BP has been used recently to establish some new information about the stable homotopy groups of spheres [9; 11]. The best results have been achieved by using the associated homology theory BP* ( ), the Hopf algebra BP*(BP), and the Adams-Novikov spectral sequence

On the differentials in the Adams spectral sequence

1964 ◽  
Vol 60 (3) ◽  
pp. 409-420 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Higher Order ◽  
Adams Spectral Sequence ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Cohomology Operations ◽  
Image Position

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements ofcorresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

scholarly journals Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence

2011 ◽  
Vol 7 (2) ◽  
pp. 203-347 ◽  
Author(s):  
Hans-Joachim Baues ◽  
Mamuka Jibladze
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Explicit Formulae ◽  
Cohomology Operations

AbstractWe 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.

Some infinite families in the stable homotopy of spheres

1987 ◽  
Vol 101 (3) ◽  
pp. 477-485 ◽  
Author(s):  
Wen-Hsiung Lin
Keyword(s):  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Stable Homotopy Of Spheres

The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.

The non-existence of odd primary Arf invariant elements in stable homotopy

1978 ◽  
Vol 83 (3) ◽  
pp. 429-443 ◽  
Author(s):  
Douglas C. Ravenel
Keyword(s):  
Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Arf Invariant ◽  
Prime 2

In this paper we will show that certain elements of order p (p an odd prime) on the 2-line of the Adams-Novikov spectral sequence support non-trivial differentials and therefore do not detect elements in the stable homotopy groups of spheres. These elements are analogous to the so-called Arf invariant elements of order 2, hence the title. However, it is evident that the methods presented here do not extend to the prime 2.

On the differentials in the Adams spectral sequence for the stable homotopy groups of spheres. II

1965 ◽  
Vol 61 (4) ◽  
pp. 855-868 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups

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.

scholarly journals Detection of a new nontrivial family in the stable homotopy of spheres $ \mbf{\pi_{\ast}S} $

2008 ◽  
Vol 39 (1) ◽  
pp. 75-83
Author(s):  
Liu Xiugui ◽  
Jin Yinglong
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Nontrivial Element ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Stable Homotopy Of Spheres ◽  
Gamma S

To determine the stable homotopy groups of spheres is one of the central problems in homotopy theory. Let $ A $ be the mod $ p $ Steenrod algebra and $S$ the sphere spectrum localized at an odd prime $ p $. In this article, it is proved that for $ p\geqslant 7 $, $ n\geqslant 4 $ and $ 3\leqslant s $, $ b_0 h_1 h_n \tilde{\gamma}_{s} \in Ext_A^{s+4,\ast}(\mathbb{Z}_p,\mathbb{Z}_p) $ is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element of order $ p $ in the stable homotopy groups of spheres $ \pi_{p^nq+sp^{2}q+(s+1)pq+(s-2)q-7}S $, where $ q=2(p-1 ) $.

On the differentials in the Adams spectral sequence for the stable homotopy groups of spheres. I

1965 ◽  
Vol 61 (1) ◽  
pp. 53-60 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Higher Order ◽  
Adams Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Cohomology Operations

In (8), we have shown how, in general, the problem of identifying the differentials in the Adams spectral sequence (see(1),(4)) is equivalent to that of calculating certain higher-order cohomology operations (in the sense of (6)). However, we propose to investigate here a slightly different method, based on the naturality of the spectral sequence, which can be used to show that certain elements are never boundaries, for any differential.

The Kahn–Priddy theorem

1973 ◽  
Vol 73 (1) ◽  
pp. 45-55 ◽  
Author(s):  
J. F. Adams
Keyword(s):  
Orthogonal Group ◽  
Homotopy Group ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Infinite Dimensional ◽  
Stable Homotopy Groups ◽  
Stable Homotopy Group ◽  
Image Position

Let ΦSr(X) be the stable homotopy groupwhere SnX means the n-fold suspension of X. For example, the groups ΦSr(S0) are the stable homotopy groups of spheres. Letbe the ‘infinite-dimensional’ orthogonal group. Then topologists are familiar with the ‘stable J-homomorphism’G. W. Whitehead observed that J factors through an ‘even more stable’ J-homomorphismhe conjectured that J′ is epi (for r > 0).

scholarly journals The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime

2012 ◽  
Vol 207 ◽  
pp. 139-157
Author(s):  
Ryo Kato ◽  
Katsumi Shimomura
Keyword(s):  
Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Spectral Sequences ◽  
First Line ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Ext Groups ◽  
Bockstein Spectral Sequence

AbstractThe chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the E2-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The E1-term of the spectral sequence is an Ext group of BP*BP-comodules. There is a sequence of Ext groups for nonnegative integers n with and there are Bockstein spectral sequences computing a module (n – s) from So far, a small number of the E1-terms are determined. Here, we determine the for p > 2 and n > 3 by computing the Bockstein spectral sequence with E1-term for s = 1, 2. As an application, we study the nontriviality of the action of α1 and β1 in the homotopy groups of the second Smith-Toda spectrum V(2).

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