The v1-periodic unstable Novikov spectral sequence

Bendersky, Martin

doi:10.1016/0040-9383(92)90063-n

Cited by 13 publications

(9 citation statements)

References 17 publications

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“…The primary difference is that those presented E 2 , while these present E ∞ when n ≡ 0, 3 mod 4, and a subquotient of E 3 after most of the d 3 -differentials have been taken into account when n ≡ 1, 2 mod 4. That the charts of [7] and [3] were of the v 1 -periodic UNSS while the ones here are of the BTSS is inconsequential, since the two spectral sequences are isomorphic in dimension > 2n + 1.…”

Section: The Localization πmentioning

confidence: 99%

Compositions in the ν1 -periodic homotopy groups of spheres

Bendersky¹,

Davis²

2003

Forum Mathematicum

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Let ρ i ∈ π n+8i−1 (S n) denote an element which suspends to a generator of the image of the stable J-homomorphism. We determine the image of the composite ρ j • ρ k in v 1-periodic homotopy v −1 1 π n+8i+8j−2 (S n). The method is to use Adams operations to compute the 1-line of an unstable homotopy spectral sequence constructed by Bendersky and Thompson. 1. Main theorem The p-primary v 1-periodic homotopy groups of a space X, denoted v −1 1 π * (X; p), are a localization of the portion of π * X (p) detected by K-theory.([14]) The v 1-periodic homotopy groups of spheres contain the image of the J-homomorphism.([18]) Until the last two sections of this paper, we will deal with 2-primary homotopy theory, let ν(−) denote the exponent of 2 in an integer, and let v −1 1 π * (X) = v −1 1 π * (X; 2). We need the following known result. Proposition 1.1. i.) v −1 1 π 2n+8i−1 (S 2n+1) ≈ Z/2 min(n,ν(i)+4) ; ii.) there are morphisms v −1 1 : π * (S 2n+1) → v −1 1 π * (S 2n+1), * > 2n + 1, which are split surjections for * = 2n + 8i − 1 with 4i − ν(i) ≥ n + 8. Proof. i.) See Theorem 2.1 and the accompanying diagrams, 2.2 and 2.3. (ii.) In [14, 1.7] and [12, 2.4] it is shown that if p e : Ω n X → Ω n X is null homotopic, then there is a natural morphism v −1 1 : π j (X) → v −1 1 π j (X) for j > n. In [22], it is noted that 2 3n/2+1 is null homotopic on Ω 2n S 2n+1 2n + 1. Thus v −1 1 is defined on π * (S 2n+1) for * > 2n + 1. The split surjection follows from [18, 1.3,1.5].

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Section: The Localization πmentioning

confidence: 99%

Compositions in the ν1 -periodic homotopy groups of spheres

Bendersky¹,

Davis²

2003

Forum Mathematicum

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“…The fibration above induces a long exact sequence in E 1,2k+1 2 (SU (n); E(1))-terms. It is this same long exact sequence and Theorem 1.4 of [5] (which relies on [4]), and Theorem 5.2 of [2] that allows the computation of the remainder of the E(1)-based E 2 -term.…”

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confidence: 94%

“…In order to calculate the desired E 2 -term, we must consider the results of Bendersky [4], and Davis [5] who compute the odd-primary v 1 -periodic homotopy groups of SU (n), denoted v −1 1 π 2k (SU (n)). These groups have been studied extensively, and are known quite well for k sufficiently large and n within a certain range.…”

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confidence: 99%

Divisible groups in the K-theory completion of SU(n)

Gregorÿ

2015

Topology and its Applications

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Results of Bendersky and Thompson for the E(1)-based E 2 -term of S 2n+1 , and the results of Bendersky and Davis concerning the v 1 -periodic groups of SU (n) are used to compute the E(1)-based E 2 -term for SU (n) at odd primes. This computation is performed using the Bendersky Thompson spectral sequence for SU (n). For spaces like SU (n) this spectral sequence converges to homotopy groups of the K-theory completion of SU (n). Of particular interest, is the existence of infinitely many divisible groups in the homotopy groups of the K-theory completion of SU which offers an example of how E-completion does not commute with direct limits.To obtain information about the homotopy of a space, it is a standard practice to use a BousfieldKan (unstable Adams) spectral sequence (BKSS). An unstable Adams spectral sequence is a sequence of functors which take as input the homology of a space, X, and under certain conditions, to be discussed below, converges to the homotopy groups of a space. In [1], an unstable Adams spectral sequence based on a generalized connective homology theory, E, was constructed, and for E = BP the homotopy groups of an odd dimensional sphere were given by the E 2 -term, E s,t 2 (S 2n+1 ; BP). This idea was extended to nonconnective periodic theories in [2] with the construction of the unstable Adams spectral sequence known as the Bendersky-Thompson spectral sequence (BTSS). In the non-connective case, however, and for the

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“…For n = 1 and Y is a mod p i Moore space, these groups have been studied by Mahowald [Mah82] and Thompson [Tho90] for X a sphere, and by Bendersky, Davis , Mahowald and Mimura ( [Ben92], [BD92], [BDM], [Dav91] and [DM92]) for X a Lie group.…”

Section: There Are Positive Integers I and J Such That The Following mentioning

confidence: 99%

Life after the Telescope Conjecture

Ravenel

1993

Algebraic K-Theory and Algebraic Topology

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We discuss the chromatic filtration in stable homotopy theory and its connections with algebraic K-theory, specifically with some results of Thomason, Mitchell, Waldhausen and McClure-Staffeldt. We offer a new definition (suggested by the failure of the telescope conjecture) of the chromatic filtration, in which all of the localization functors used are finite.The telescope conjecture (2.2 below) was first announced by the author in 1977, published in [Rav84], and disproved in 1990 [Rav92b] [Rav]. It is a statement about the chromatic filtration, which was originally devised to understand the stable homotopy theory of finite complexes. Very briefly, for each prime there is a series of localization functors L n defined in the homotopy category (both stably and unstably) so that for each spectrum X there is an inverse system (called the chromatic tower)The chromatic filtration of π * (X) defined in terms of the kernels of the mapsThe chromatic convergence theorem (1.2) says that the inverse limit is X when X is a p-local finitecomplex. L 0 X is the rational homotopy type of X and L 1 X is its localization with respect to p-local complex K-theory. Both of these functors are well understood. The other functors are less familiar but manageable, and we have a good handle on the n th 'layer' of the chromatic tower, i.e., the fibre of the map

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The v1-periodic unstable Novikov spectral sequence

Cited by 13 publications

References 17 publications

Compositions in the ν1 -periodic homotopy groups of spheres

Compositions in the ν1 -periodic homotopy groups of spheres

Divisible groups in the K-theory completion of SU(n)

Life after the Telescope Conjecture

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