On the existence of the self map 𝑣⁹₂ on the Smith-Toda complex 𝑉(1) at the prime 3

Behrens, Mark; Pemmaraju, Satya

doi:10.1090/conm/346/06284

Cited by 8 publications

(8 citation statements)

References 16 publications

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“…In [8], we proved the 'only if' part. For the survivors, we have β s for s > 0 with s ≡ 0, 1, 2, 5, 6 mod 9 by M. Behrens and S. Pemmaraju [1].…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Note on beta elements in homotopy, and an application to the prime three case

Shimomura

2009

Proc. Amer. Math. Soc.

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Abstract. Let S 0 (p) denote the sphere spectrum localized at an odd prime p. Then we have the first beta element β 1 ∈ π 2p 2 −2p−2 (S 0 (p) ), whose cofiber we denote by W . We also consider a generalized Smith-Toda spectrum V r characterized by BP * (V r ) = BP * /(p, v r 1 ). In this note, we show that an element of π * (V r ∧ W ) gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of β 9t+3 at the prime three to complete a conjecture of Ravenel's: β s ∈ π 16s−6 (S 0 (3) ) exists if and only if s is not congruent to 4, 7 or 8 mod 9.

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“…In [8], we proved the 'only if' part. For the survivors, we have β s for s > 0 with s ≡ 0, 1, 2, 5, 6 mod 9 by M. Behrens and S. Pemmaraju [1].…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Note on beta elements in homotopy, and an application to the prime three case

Shimomura

2009

Proc. Amer. Math. Soc.

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“…The identity map V (1) → V (1) does have order 3. By [6], V (1) has a v 9 2 -self map, and hence L K(2) V (1) is 144-periodic. In [12] we showed that the homotopy groups of L K(2) V (1) are exactly 144-periodic; see…”

Section: The Duality Theoremmentioning

confidence: 99%

Comparing Dualities in the K(n)-local Category

Goerss¹,

Hopkins²

2021

Equivariant Topology and Derived Algebra

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In their work on the period map and the dualizing sheaf for Lubin-Tate space, Gross and the second author wrote down an equivalence between the Spanier-Whitehead and Brown-Comenetz duals of certain type ncomplexes in the K(n)-local category at large primes. In the culture of the time, these results were accessible to educated readers, but this seems no longer to be the case; therefore, in this note we give the details. Because we are at large primes, the key result is algebraic: in the Picard group of Lubin-Tate space, two important invertible sheaves become isomorphic modulo p.For John Greenlees, the master of duality. P. G. Goerss and M. J. Hopkins K(n)-local category is that under some circumstances the two dualities are closely related.Recall that a finite spectrum X is of type n if K(m) * X = 0 for m < n. By [22], any type n spectrum has a v p k n -self map; that is, there is an integer k and mapthat is, for any a ∈ Z n we have a sphere S 2a . (The phrase "p-adic sphere" is common here, but misleading: Z n is not the p-adic integers. See Remark 1.23.) Now let λ = lim k p nk r(n) ∈ Z n . The key algebraic result can now be deduced from Proposition 1.30 below: under the composition

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“…The identity map V (1) → V (1) does have order 3. By [BP04], V (1) has a v 9 2 -self map, and hence L K(2) V (1) is 144-periodic. In [GHM04] we showed that the homotopy groups of L K(2) V (1) are exactly 144-periodic; see also the charts at the end of [GH16].…”

Section: The Duality Theoremmentioning

confidence: 99%