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).

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 Fold cobordisms and stable homotopy groups

2009 ◽  
Vol 46 (4) ◽  
pp. 437-447
Author(s):  
Boldizsár Kalmár
Keyword(s):  
Direct Summand ◽  
Homotopy Group ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Cobordism Group ◽  
Stable Homotopy Groups ◽  
Direct Sum ◽  
Stable Homotopy Group

We prove that for n ≧ 1 and q > 0 the (oriented) cobordism group of fold maps of (oriented) ( n + q )-dimensional manifolds into ℝ n contains the direct sum of ⌊ q + 1)/2⌋ copies of the ( n − 1)th stable homotopy group of spheres as a direct summand. We also prove that for k ≧ 1 and q = 2 k −1 the cobordism group of fold maps of unoriented ( n + q )-dimensional manifolds into ℝ n also contains the n th stable homotopy group of the space ℝ P∞ as a direct summand. We have the analogous results about bordism groups of fold maps as well.

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.

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