Composition properties of projective homotopy classes

Feder, S.; Gitler, S.; Lam, Kee Yuen

doi:10.2140/pjm.1977.68.47

Cited by 15 publications

(24 citation statements)

References 13 publications

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“…The authors thank Professor K.Y. Lam who pointed out the numerical condition ensuring the triviality of Whitehead products between ι n and elements of the stable J -image [7].…”

Section: Acknowledgementsmentioning

confidence: 99%

Gottlieb groups of spheres

Golasiński

Mukai

2008

Topology

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This paper takes up the systematic study of the Gottlieb groups G n+k (S n ) of spheres for k ≤ 13 by means of the classical homotopy theory methods. We fully determine the groups G n+k (S n ) for k ≤ 13 except for the 2-primary components in the cases: k = 9, n = 53; k = 11, n = 115. In particular, we show [ι n , η 2The Gottlieb groups G k (X ) of a pointed space X have been defined by Gottlieb in [9] and [10]; first G 1 (X ) and then G k (X ) for all k ≥ 1. The higher Gottlieb groups G k (X ) are related in [10] to the existence of sectioning fibrations with fiber X . For instance, if G k (X ) is trivial then there is a crosssection for every fibration over the (k + 1)-sphere S k+1 , with fiber X . This paper grew out of our attempt to develop techniques in calculating G n+k (S n ) for k ≤ 13 and any n ≥ 1. The composition methods developed by Toda [36] are the main tools used in the paper. Our calculations also deeply depend on the results of [13,16,21].

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“…The authors thank Professor K.Y. Lam who pointed out the numerical condition ensuring the triviality of Whitehead products between ι n and elements of the stable J -image [7].…”

Section: Acknowledgementsmentioning

confidence: 99%

Gottlieb groups of spheres

Golasiński

Mukai

2008

Topology

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“…We will use the maps <f>: Pl+\-* P£ mentioned in the introduction. They were first constructed in [8] and studied in more detail in [10]. Because <j> is defined only when / is even, Theorem 1.1 is somewhat more difficult to prove when n is odd; the case n odd will be considered in Remark 4.4.…”

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confidence: 99%

The Stable Geometric Dimension of Vector Bundles Over Real Projective Spaces

Davis¹,

Gitler²,

Mahowald³

1981

Transactions of the American Mathematical Society

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Abstract. An elementary argument shows that the geometric dimension of any vector bundle of order 2e over RP" depends only on e and the residue of n mod 8 for n sufficiently large. In this paper we calculate this geometric dimension, which is approximately le. The nonlifting results are easily obtained using the spectrum bJ. The lifting results require fto-resolutions. Half of the paper is devoted to proving Mahowald's theorem that beginning with the second stage io-resolutions act almost like ^(Z^-resolutions.

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“…and 0 ≤ λ ≤ p − 2, but this last restriction together with (6) implies that such elements lie in C(i − 1).…”

Section: Proof Of 21 the Compositementioning

confidence: 99%

Odd primary 𝑏𝑜 resolutions and classification of the stable summands of stunted lens spaces

González

1999

Trans. Amer. Math. Soc.

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Abstract. The classification of the stable homotopy types of stunted lens spaces and their stable summands can be obtained by proving the triviality of certain stable classes in the homotopy groups of these spaces. This is achieved in the 2-primary case by Davis and Mahowald using classical Adams spectral sequence techniques. We obtain the odd primary analogue using the corresponding Adams spectral sequence based at the spectrum representing odd primary connective K-theory. The methods allow us to answer a stronger problem: the determination of the smallest stunted space where such stable classes remain null homotopic. A technical problem prevents us from giving an answer in all situations; however, in a quantitative way, the number of cases missed is very small.

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Composition properties of projective homotopy classes

Cited by 15 publications

References 13 publications

Gottlieb groups of spheres

Gottlieb groups of spheres

The Stable Geometric Dimension of Vector Bundles Over Real Projective Spaces

Odd primary 𝑏𝑜 resolutions and classification of the stable summands of stunted lens spaces

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