The Double Suspension and Exponents of the Homotopy Groups of Spheres

Cohen, Fred; Moore, John C.; Neisendorfer, Joseph

doi:10.2307/1971238

Cited by 130 publications

(129 citation statements)

References 10 publications

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“…B]. Note also that, in general, this inclusion is proper as is shown in the following example comunicated to us by F. Cohen: It is known [4,Cor. 1.3] that, given a prime p ≥ 3 and n ≥ 1, p n is an exponent for S 2n+1 at p. Therefore, if we consider ρ the p n -th power map on the space X = (Ω 2n−3 S 2n+1 2n + 1 ) (p) and call σ = 1+ρ, it follows that π * (σ) = 1 π * (X) .…”

Section: Remarks (A)mentioning

confidence: 77%

A bound for the nilpotency of a group of self homotopy equivalences

Félix

Murillo²

1998

Proc. Amer. Math. Soc.

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Abstract. Let E Ω (X) be the group of homotopy classes of self-homotopy equivalences of X such that Ωf 1d ΩX . We prove that E Ω (X) is a nilpotent group and that nil E Ω (X) ≤ cat(X) − 1.Given a pointed space X of the homotopy type of a CW-complex, let E(X) denote the group of based homotopy classes of self homotopy equivalences of X ([1] is an excellent survey on this object). A considerable amount of work has been dedicated to obtaining finiteness properties, not only of E(X), but also of certain interesting subgroups which preserve additional geometrical structure (see for example [2], [5], [6], [8]). This note goes in this direction: Let E Ω (X) be the kernel of the obvious map E(X) → E(ΩX) (i.e. homotopy classes of equivalences f : X → X such that Ωf 1 ΩX ) and, as usual, denote by cat(X) the LusternikSchnirelmann category of X. Then we prove:Theorem. If cat X is finite then E Ω (X) is a nilpotent group and nil E Ω (X) ≤ cat(X) − 1.

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Section: Remarks (A)mentioning

confidence: 77%

A bound for the nilpotency of a group of self homotopy equivalences

Félix

Murillo²

1998

Proc. Amer. Math. Soc.

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“…These groups are interesting because they are often almost completely computable and yet they give large direct summands of actual homotopy groups of X. Using results of [17] and [11], we show at the end of this section that all compact Lie groups have such exponents for all p , and so we should try to compute their vxperiodic homotopy groups. This was done at the odd primes for SU(«), Sp(n), and SO(«) in [13], for SU(«) at the prime 2 in [8], and for G2 at the prime 2 in [14].…”

Section: Statement Of Resultsmentioning

confidence: 97%

“…The attaching map of the middle cell of 5(2« + 1, 2« + 2/7 -1) is ax £ n2n+2p-2(S2n+x). The only factor not of this type, labeled B2 (3,11), is a sphere bundle with attaching map a2, and is not a direct factor in the ^-localization of a quotient of SU 's.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…It was proved in [11] that expp(5'2"+1) = « , and in particular does not depend upon p. If vx~xn*(X) has an element of order pe, then so does n*(X), although not necessarily for the same value of * . Since the p-exponent of the total space of a fibration is < the sum of the /^-exponents of the base and the fiber, and since the /^-exponent of a product is the maximum of the /7-exponents of its factors, we have the following corollary of the above results.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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𝑣₁-periodic homotopy groups of exceptional Lie groups: torsion-free cases

Bendersky¹,

Davis²,

Mimura³

1992

Trans. Amer. Math. Soc.

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“…This fits with a conjecture of Barratt which says that if the identity map of a double suspension has order p r then its homotopy groups should have exponent p r C1 . As a very important application [3], they also used their decomposition of P 2nC1 .p r / to construct a map W 2 S 2nC1 ! S 2n 1 having the property that the composite E 2 ı W 2 S 2nC1 !…”

Section: Introductionmentioning

confidence: 99%

The decomposition of the loop space of the mod 2 Moore space

Grbić

Selick

2008

Algebr. Geom. Topol.

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In 1979 Cohen, Moore and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod p Moore space for primes p > 2 and used the results to find the best possible exponent for the homotopy groups of spheres and for Moore spaces at such primes. The corresponding problems for p D 2 are still open. In this paper we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod 2 Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod 2 Moore space or to an improvement in the known bounds for the exponent of the 2-torsion in the homotopy groups of spheres.

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The Double Suspension and Exponents of the Homotopy Groups of Spheres

Cited by 130 publications

References 10 publications

A bound for the nilpotency of a group of self homotopy equivalences

A bound for the nilpotency of a group of self homotopy equivalences

𝑣₁-periodic homotopy groups of exceptional Lie groups: torsion-free cases

The decomposition of the loop space of the mod 2 Moore space

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