scholarly journals The cohomology of the Steenrod algebra; stable homotopy groups of spheres

1965 ◽  
Vol 71 (2) ◽  
pp. 377-381 ◽  
Author(s):  
J. Peter May
Keyword(s):  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups

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 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 ) $.

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 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

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