The triviality of the 61-stem in the stable homotopy groups of spheres

Wang, Guozhen; Xu, Zhouli

doi:10.4007/annals.2017.186.2.3

Cited by 36 publications

(48 citation statements)

References 46 publications

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“…This includes a verification of the details in the classical literature [2] [3] [4] [25]. Subsequently, the second and third authors computed the 60-stem and 61-stem [38]. We also mention [21] [22], which take an entirely different approach to computing stable homotopy groups.…”

Section: Introductionmentioning

confidence: 96%

Stable Stems

Isaksen

2019

Memoirs of the AMS

104

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We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, η, and ν, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences.We also compute the motivic stable homotopy groups of the cofiber of the motivic element τ . This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of τ are the same as the E 2 -page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known. ContentsList of Tables vii Chapter 1. Introduction 1.1. The Adams spectral sequence program 1.2. Motivic homotopy theory 1.3. The motivic Steenrod algebra 1.4. Relationship between motivic and classical calculations 1.5. Relationship to the Adams-Novikov spectral sequence 1.6. How to use this manuscript 1.7. Notation 1.8. Acknowledgements Chapter 2. The cohomology of the motivic Steenrod algebra 2.1. The motivic May spectral sequence 2.2. Massey products in the motivic May spectral sequence 2.3. The May differentials 2.4. Hidden May extensions Chapter 3. Differentials in the Adams spectral sequence 3.1. Toda brackets in the motivic Adams spectral sequence 3.2. Adams differentials 3.3. Adams differentials computations Chapter 4. Hidden extensions in the Adams spectral sequence 4.1. Hidden Adams extensions 4.2. Hidden Adams extensions computations Chapter 5. The cofiber of τ 5.1. The Adams E 2 -page for the cofiber of τ 5.2. Adams differentials for the cofiber of τ 5.3. Hidden Adams extensions for the cofiber of τ Chapter 6. Reverse engineering the Adams-Novikov spectral sequence 6.1. The motivic Adams-Novikov spectral sequence 6.2. The motivic Adams-Novikov spectral sequence for the cofiber of τ 6.3. Adams-Novikov calculations Chapter 7. Tables Index Bibliography v 45 Hidden Adams-Novikov η extensions 46 Hidden Adams-Novikov ν extensions 47 Correspondence between classical Adams and Adams-Novikov E ∞ 48 Classical Adams-Novikov boundaries 49 Classical Adams-Novikov non-permanent classes

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Section: Introductionmentioning

confidence: 96%

Stable Stems

Isaksen

2019

Memoirs of the AMS

104

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“…It is very helpful when thinking of CW spectra. See [1,20,22] for example. For simplicity, we restrict to the cases when the spectra X, Y, Z, W are all spheres.…”

Section: Appendixmentioning

confidence: 99%

“…We also show the two statements in Proposition 1.3 are equivalent. In Section 3, we recall a few notations from [20]. We also give a brief review of how to use the RP ∞ technique to prove differentials and to solve extension problems.…”

Section: Introductionmentioning

confidence: 99%

Some extensions in the Adams spectral sequence and the 51–stem

Wang

2018

Algebr. Geom. Topol.

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We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of π 51 is Z/8 ⊕ Z/8 ⊕ Z/2. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors ([5], [7], [20]) through the 61-stem.The proof of this result uses the RP ∞ technique, which was introduced by the authors in [20] to prove π 61 = 0. This paper advertises this technique through examples that have simpler proofs than in [20].Proposition 1.1. There is a nontrivial 2-extension from h 0 h 3 g 2 to gn in the 51-stem. the 51-stem and some extensionsWe first establish the following lemma. Lemma 2.1. In the Adams E 2 -page, we have the following Massey products in the 46-stem: gn = N, h 1 , h 2 = N, h 2 , h 1 Proof. By Bruner's computation [4], there is a relation in bidegree (t − s, s) = (81, 15): gnr = mN.

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“…If the dimension d of M is such that the d-sphere admits a unique differentiable structure, then it is possible to choose N = M (see Theorem 3.1). It is known that the differentiable structure on S d is unique for d ∈ {1, 2, 3, 5, 6, 12, 56, 61} [24,28,42], and never unique for odd d / ∈ {1, 3, 5, 61} [24,42], while a general answer is still unknown in even dimension (and is a wide open problem for d = 4). Remark 1.3.…”

Section: Introductionmentioning

confidence: 99%

Property FW, differentiable structures and smoothability of singular actions

Lodha

Bon

Triestino

2020

Journal of Topology

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We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group normalΓ has the fixed point property FW for walls (for example, if it has property (T)), every aperiodic action of normalΓ by diffeomorphisms that are of class Cr with countably many singularities is conjugate to an action by true diffeomorphisms of class Cr on a homeomorphic (possibly non‐diffeomorphic) manifold. As applications, we show that Navas's result for actions of Kazhdan groups on the circle, as well as the recent solutions to Zimmer's conjecture, generalise to aperiodic actions by diffeomorphisms with countably many singularities.

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The triviality of the 61-stem in the stable homotopy groups of spheres

Cited by 36 publications

References 46 publications

Stable Stems

Stable Stems

Some extensions in the Adams spectral sequence and the 51–stem

Property FW, differentiable structures and smoothability of singular actions

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