Periodic Phenomena in the Adams-Novikov Spectral Sequence

Miller, Haynes; Ravenel, Douglas C.; Wilson, W. Stephen

doi:10.2307/1971064

Cited by 230 publications

(185 citation statements)

References 18 publications

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“…for example [GHMR05]. We have E 0, * 2 = E 0,0 2 = Z, the groups E 1,t 2 are finite cyclic with order given by denominators of Bernoulli numbers, and E 2, * 2 is very complicated but known explicitly by [MRW77]. A conceptual interpretation of E 2, * 2 in terms of congruences between elliptic modular forms was only achieved recently [BL] using the topological modular forms of Goerss, Hopkins and Miller.…”

Section: Introductionmentioning

confidence: 99%

The f-invariant and index theory

Bunke

Naumann

2010

manuscripta math.

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Section: Introductionmentioning

confidence: 99%

The f-invariant and index theory

Bunke

Naumann

2010

manuscripta math.

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“…Then the semi-cosimplicial-graded abelian group is of the form [15]. Thus, below Ell * denotes p-local elliptic homology (p ≥ 5), and a p-typical orientation BP * → Ell * is fixed, as in [5].…”

Section: 2mentioning

confidence: 99%

ON THE DOUBLE TRANSFER AND THE f-INVARIANT

Powell¹

2012

Glasgow Math. J.

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Abstract. The purpose of this paper is to investigate the algebraic double S 1 -transfer, in particular the classes in the two-line of the Adams-Novikov spectral sequence which are the image of comodule primitives of the MU-homology of ‫ރ‬P ∞ × ‫ރ‬P ∞ via the algebraic double transfer. These classes are analysed by two related approaches: the first, p-locally for p ≥ 3, by using the morphism induced in MUhomology by the chromatic factorisation of the double transfer map together with the f -invariant of Behrens (for p ≥ 5) (M. Behrens, Congruences between modular forms given by the divided β-family in homotopy theory, Geom. Topol. 13(1) (2009), 319-357). The second approach (after inverting 6) uses the algebraic double transfer and the f -invariant of Laures (G. Laures, The topological q-expansion principle, Topology 38(2) (1999), 387-425).

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“…In [7], the β-element β s of the E 2 -term E (S 0 ) associated to the cofiber sequences of (1.1). It is shown [13] that the β-element β s ∈ E 2,16s−4 2 (S 0 ) survives to a homotopy element of π 16s−6 (L 2 S 0 ) if s ≡ 0, 1, 2, 3, 5, 6 mod 9, which corresponds to one of Ravenel's conjectures in π * (S 0 ).…”

Section: Lemma 15 Let U Be An Invertible Spectrum With E(2)mentioning

confidence: 99%

𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3

Ichigi

Shimomura

2004

Proc. Amer. Math. Soc.

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Abstract. Let L 2 denote the Bousfield localization functor with respect to the Johnson-Hovey and Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E(2)-local sphere L 2 S 0 at a prime p > 3. At the prime 3, it is shown, A relation between the Picard group of the E(n)-local homotopy category and E(n)-based Adams spectral sequence, that there exists an invertible spectrum X that is not homotopy equivalent to a suspension of L 2 S 0 . In this paper, we show the homotopy equivalence v 3 2 : Σ 48 L 2 V (1) V (1) ∧ X for the Smith-Toda spectrum V (1). In the same manner as this, we also show the existence of the self-map β : Σ 144 L 2 V (1) → L 2 V (1) that induces v 9 2 on the E(2) * -homology.

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Periodic Phenomena in the Adams-Novikov Spectral Sequence

Cited by 230 publications

References 18 publications

The f-invariant and index theory

The f-invariant and index theory

ON THE DOUBLE TRANSFER AND THE f-INVARIANT

𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3

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