The iterated transfer analogue of the new doomsday conjecture

Minami, Norihiko

doi:10.1090/s0002-9947-99-02037-1

Cited by 52 publications

(56 citation statements)

References 33 publications

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“…The Conner-Floyd conjecture was proved in the early 80's: [29,32], and it turns out that, together with detailed information about the p-series, the above description of I n leads in fact to a full description of the (additive) structure of the Brown-Peterson homology of (BZ/ p) ∧n [20,21]. The relevance of such a calculation has been confirmed by Minami's work [25][26][27][28] on the possible existence of framed manifolds of Kervaire invariant 1 (that is, on the basic problem of understanding stable homotopy classes of spheres detected in the 2-line of the classical Adams spectral sequence). Now, in view of the basic role the p-series played in the above development, it would be interesting to see to what extent the information in this paper for the p k -series can be used in a calculation of BP * (BZ/ p k 1 × • • • × BZ/ p k n ), as well as its implications in the stable homotopy groups of spheres.…”

Section: Appendixmentioning

confidence: 84%

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González

2003

Journal of Algebraic Combinatorics

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The 1-dimensional universal formal group law is a power series (in two variables and with coefficients in Lazard's ring) carrying a lot of geometrical and algebraic properties. For a prime p, we study the corresponding " p-localized" formal group law through its associated p k-series, [ p k ](x) = s≥0 a k,s x s(p−1)+1-the p k-fold iterated formal sum of a variable x. The coefficients a k,s lie in the Brown-Peterson ring BP * = Z (p) [v 1 , v 2 ,. . .] and we describe part of their structure as polynomials in the variables v i with p-local coefficients. This is achieved by introducing a family of filtrations {W ϕ } ϕ≥1 in BP * and studying the value of a k,s in each of the associated (bi)graded rings BP * /W ϕ. This allows us to identify, among monomials in a k,s of minimal W ϕ-filtration (1 ≤ ϕ ≤ k), an explicit monomial m ϕ,k,s carrying the lowest possible p-divisibility. The p-local coefficient of m ϕ,k,s is described as a Stirling-type number of the second kind and its actual value is computed up to p-local units. It turns out that m k,k,s not only carries the lowest W k-filtration but, more importantly, the lowest p-divisibility among all other monomials in a k,s. In particular, we obtain a complete description of the p-divisibility properties of each a k,s .

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Section: Appendixmentioning

confidence: 84%

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González

2003

Journal of Algebraic Combinatorics

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“…2 ) be the subspace of H * (F s 2 ) consisting of all elements that are annihilated by all positive degree Steenrod squares, then there is an induced action of GL s on P H * (F s 2 ), and we have an F 2 -linear map from the coinvariant elements of P H * (F s 2 ) to mod-2 cohomology group Ext s,s+ * A (F 2 , F 2 ) of the Steenrod algebra, which is induced over the E 2 -term of the Adams spectral sequence by the geometrical transfer map Σ ∞ (B(F s 2 ) + ) −→ Σ ∞ (S 0 ) in stable homotopy theory (see also Mitchell [18]). These transfers can be played a key role in the study of the Kervaire invariant one problem (see Browder [3], Mahowald [15], Minami [16,17]). The Kervaire invariant was first introduced by Browder's work [3], where he shows that the classes h 2 j ∈ Ext 2,2 j+1 A (F 2 , F 2 ) are the permanent cycles in the classical Adams spectral sequence at the prime 2, if and only if smooth framed manifolds of Kervaire invariant one exist only in dimensions 2 j+1 − 2.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We explicitly compute p (i;I) (S) in terms u j , 1 j 23. By a direct computation using Lemma 2.2.9,Theorem 3.1.3, and from the relations p (i;j) (S) ≡ ω (5,2) 0 with either i = 1, j = 2, 3 or i = 2, j = 3, 4, one gets Here J = {1, 2, 3,4,5,6,7,8,9,10,11,12,15,16,17,18,19,21,22,23,24,25,26,28,29,31,38,39,40,41,42,45,46,48,50,51,57 [46]). By a simple computation, we get…”

Section: We Now Prove the Setmentioning

confidence: 99%

The "hit" problem of five variables in the generic degree and its application

Phúc¹

2021

Preprint

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Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree $1$. We are interested in the {\it Peterson "hit" problem} of finding a minimal set of generators for $P_s$ as a graded left module over the mod-2 Steenrod algebra, $\mathscr {A}$. For $s\geqslant 5,$ it is still open.In this paper, we study the hit problem of five variables in a generic degree. By using this result, we survey Singer's conjecture for the fifth algebraic transfer in the respective degrees. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer

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“…the Kameko squaring operation. This was shown by Boardman [2] and Minami [26] to commute with the classical squaring operation through the algebraic transfer T r s :…”

Section: Preliminarymentioning

confidence: 95%

The squaring operation on -generators of the Dickson algebra.

Hùng¹,

Quỳnh²

2009

Math. Proc. Camb. Phil. Soc.

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We study the squaring operation Sq 0 on the dual of the minimal A-generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes-Zarati homomorphism vanishes (1) on every element in any finite Sq 0 -family in E xt * A (F 2 , F 2 ) except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes. Dedicated to William Singer on the occasion of his 65th birthdayfor any s (see [15]). Therefore the investigation of the squaring operation is useful to the study of the Lannes-Zarati homomorphism. † Supported in part by a grant of the NAFOSTED. NGUYỄN H. V. HƯNG AND VÕ T. N. QUỲNHThe Lannes-Zarati homomorphism, defined in [23], is the one corresponding to an associated graded of the Hurewicz map H : π s * (S 0 ) % π * (Q 0 S 0 ) → H * (Q 0 S 0 ). So, the following is an algebraic version of the conjecture on spherical classes. CONJECTURE 1·1 ([12]). ϕ s = 0 in any positive stem for s > 2.That the conjecture is no longer valid for s = 1 and 2 is respectively an exposition of the existence of Hopf invariant one and Kervaire invariant one classes. (See Adams [1], Browder [4], Curtis [7], Snaith and Tornehave [31], Wellington [34] for a discussion on spherical classes; and see Lannes-Zarati [23], Goerss [10], Hưng [12]-[15] for a discussion on the homomorphism.)The squaring operation on P(F 2 ⊗

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The iterated transfer analogue of the new doomsday conjecture

Cited by 52 publications

References 33 publications

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The "hit" problem of five variables in the generic degree and its application

The squaring operation on -generators of the Dickson algebra.

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