The homotopy groups π∗(L2S0) at the prime 3

Shimomura, Katsumi; Wang, Xiangjun

doi:10.1016/s0040-9383(01)00033-7

Cited by 28 publications

(25 citation statements)

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“…As mentioned above, this result can be used to organize the already existing and very complicated calculations of Shimomura ([24], [25]) and it also suggests an independent approach to these calculations. Other applications would be to the study of Hopkins's Picard group (see [12]…”

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confidence: 86%

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A resolution of the K(2)-local sphere at the prime 3

et al. 2005

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We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L K(2) S 0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E hF 2 where F is a finite subgroup of the Morava stabilizer group and E 2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n = 2 at p = 3 represents the edge of our current knowledge: n = 1 is classical and at n = 2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic.The problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics. A period of calculation beginning with Serre's computation of the cohomology of Eilenberg-MacLane spaces and the advent of the Adams spectral sequence culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on periodic phenomena in the homotopy groups of spheres and Ravenel's nilpotence conjectures. The solutions to most of these conjectures by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of the "chromatic" point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory. Computations passed from center stage, to some extent, although there has been steady work in the wings

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confidence: 86%

“…(See, for example, [23], [24] and [25].) One aim of this paper is to provide a conceptual framework for organizing those results and produce further advances.…”

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confidence: 99%

A resolution of the K(2)-local sphere at the prime 3

et al. 2005

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“…The case p ≥ 5 is treated in [SY95] and is also nicely presented in [Beh12]. The case p = 3 is treated in [SW02b,Shi00] and the case p = 2 is partially treated in [SW02a,Shi99].…”

Section: Introductionmentioning

confidence: 95%

The 𝛼-family in the 𝐾(2)-local sphere at the prime 2

Beaudry¹

2019

Homotopy Theory: Tools And Applications

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“…Indeed, Behrens [4] constructs Q (2) in an effort to reinterpret previous groundbreaking work [5][6][7] on π * L K(2) S (3) -groups which lie at the edge of what is accessible computationally, as very little is known about the homotopy of the K(n)-local sphere at any prime for n 3. The spectrum Q (2) is an E ∞ ring spectrum with the property that…”

Section: Introductionmentioning

confidence: 99%

The Adams–Novikov E2-term for Behrens' spectrum Q(2) at the prime 3

Larson

2015

Journal of Pure and Applied Algebra

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The homotopy groups π∗(L2S0) at the prime 3

Cited by 28 publications

References 9 publications

A resolution of the K(2)-local sphere at the prime 3

A resolution of the K(2)-local sphere at the prime 3

The 𝛼-family in the 𝐾(2)-local sphere at the prime 2

The Adams–Novikov E2-term for Behrens' spectrum Q(2) at the prime 3

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