Two-primary analogues of Selick's theorem and the Kahn-Priddy theorem for the 3-sphere

Cohen, Fred

doi:10.1016/0040-9383(84)90003-x

Cited by 16 publications

(12 citation statements)

References 13 publications

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“…According to [4], Ω 2 (S 3 3 ) has an exponent 4 and hence the result. By Proposition 2.23, there are Z/8-summands in π * (S n {2}) for n = 4, 8, 16, 32 or 64.…”

Section: 52mentioning

confidence: 97%

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Homotopy theory of the suspensions of the projective plane

Wu¹

2003

Memoirs of the AMS

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“…According to [4], Ω 2 (S 3 3 ) has an exponent 4 and hence the result. By Proposition 2.23, there are Z/8-summands in π * (S n {2}) for n = 4, 8, 16, 32 or 64.…”

Section: 52mentioning

confidence: 97%

“…It is unknown whether the Barratt conjecture holds for P n (2 r ) with r ≥ 2. But there is a family of Z/2 r+1 -summands in π * (P n (2 r )), see [4], and so 2 r+1 will be the best exponent if the Barratt conjecture holds.…”

Section: Spherical Fibrationsmentioning

confidence: 99%

Homotopy theory of the suspensions of the projective plane

Wu¹

2003

Memoirs of the AMS

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“…Explicit decompositions of Ω 2 S 5 {2}, Ω 2 S 9 {2} and Ω 3 S 17 {2} corresponding to the first three 2primary Kervaire invariant classes θ 1 = η 2 , θ 2 = ν 2 and θ 3 = σ 2 are given in [5], [9] and [1].…”

Section: Introductionmentioning

confidence: 99%

The fibre of the degree $3$ map, Anick spaces and the double suspension

Amelotte¹

2019

Preprint

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Let S 2n+1 {p} denote the homotopy fibre of the degree p self map of S 2n+1 . For primes p ≥ 5, work of Selick shows that S 2n+1 {p} admits a nontrivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a nontrivial decomposition of ΩS 2n+1 {p} implies the existence of a p-primary Kervaire invariant one element of order p in π S 2n(p−1)−2 . We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS 2n+1 {p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ 3 to give a new decomposition of ΩS 55 {3} analogous to Selick's decomposition of ΩS 2p+1 {p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension S 2n−1 −→ Ω 2 S 2n+1 is homotopy equivalent to the double loop space of Anick's space.

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“…The following homological result was proved in [1] and used to obtain the homotopy decompositions of [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

A homotopy decomposition of the fibre of the squaring map on $\Omega^3 S^{17}$

Amelotte

2018

Homology, Homotopy and Applications

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We use Richter's 2-primary proof of Gray's conjecture to give a homotopy decomposition of the fibre Ω 3 S 17 {2} of the H-space squaring map on the triple loop space of the 17-sphere. This induces a splitting of the mod-2 homotopy groups π * (S 17 ; Z/2Z) in terms of the integral homotopy groups of the fibre of the double suspension E 2 : S 2n−1 → Ω 2 S 2n+1 and refines a result of Cohen and Selick, who gave similar decompositions for S 5 and S 9 . We relate these decompositions to various Whitehead products in the homotopy groups of mod-2 Moore spaces and Stiefel manifolds to show that the Whitehead square [i 2n , i 2n ] of the inclusion of the bottom cell of the Moore space P 2n+1 (2) is divisible by 2 if and only if 2n = 2, 4, 8 or 16. This will follow from a preliminary result (Proposition 3.1) equating the divisibility of [i 2n , i 2n ] with the vanishing of a Whitehead product in the mod-2 homotopy of the Stiefel manifold V 2n+1,2 , i.e., the unit tangent bundle over S 2n . It is shown in [17] that there do not exist maps S 2n−1 × P 2n (2) → V 2n+1,2 extending the inclusions of the bottom cell S 2n−1 and bottom Moore space P 2n (2) if 2n = 2, 4, 8 or 16.When 2n = 2, 4 or 8, the Whitehead product obstructing an extension is known to vanish for reasons related to Hopf invariant one, leaving only the boundary case 2n = 16 unresolved. We find that the Whitehead product is also trivial in this case.2. The decomposition of Ω 3 S 17 {2}

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Two-primary analogues of Selick's theorem and the Kahn-Priddy theorem for the 3-sphere

Cited by 16 publications

References 13 publications

Homotopy theory of the suspensions of the projective plane

Homotopy theory of the suspensions of the projective plane

The fibre of the degree $3$ map, Anick spaces and the double suspension

A homotopy decomposition of the fibre of the squaring map on $\Omega^3 S^{17}$

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