On relations between Adams spectral sequences, with an application to the stable homotopy of a moore space

Miller, Haynes

doi:10.1016/0022-4049(81)90064-5

Cited by 111 publications

(76 citation statements)

References 19 publications

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“…A more detailed exposition can be found in [9], see also [10] and [18]. The original reference for Adams resolutions (in the classical sense) is [26].…”

Section: Adams Resolutionsmentioning

confidence: 99%

I–adic towers in topology

Wüthrich

2005

Algebr. Geom. Topol.

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A large variety of cohomology theories is derived from complex cobordism M U * (−) by localizing with respect to certain elements or by killing regular sequences in M U * . We study the relationship between certain pairs of such theories which differ by a regular sequence, by constructing topological analogues of algebraic I -adic towers. These give rise to Higher Bockstein spectral sequences, which turn out to be Adams spectral sequences in an appropriate sense. Particular attention is paid to the case of completed Johnson-Wilson theory E(n) and Morava K -theory K(n) for a given prime p.

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“…A more detailed exposition can be found in [9], see also [10] and [18]. The original reference for Adams resolutions (in the classical sense) is [26].…”

Section: Adams Resolutionsmentioning

confidence: 99%

I–adic towers in topology

Wüthrich

2005

Algebr. Geom. Topol.

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“…The homotopy groups π * (L 2 X ∧ V (1)) are isomorphic to the tensor product of the exterior algebra Λ(ζ 2 ) and the K(2) * -module generated by b 0 , b 5 , b 6 , b 10 , b 11 and b 15 for the generators b l ∈ π l (L 2 X ∧ V (1)). This is used to determine the E 2 -term of the Adams-Novikov spectral sequence for computing π * (L 2 V (1)) (see Theorem 5.8).…”

Section: Theorem Bmentioning

confidence: 99%

“…Theorem C. The homotopy groups π * (L 2 V 1 ) are isomorphic to the tensor product of the exterior algebra Λ(ζ 2 ) and the K(2) * -module generated by a 0 , a 3 , a 6 , a 11 , a 13 , a 21 , a 34 and a 40 for the generators a l ∈ π l (L 2 V 1 ).…”

Section: Theorem Bmentioning

confidence: 99%

The homotopy groups of the 𝐿₂-localized Toda-Smith spectrum 𝑉(1) at the prime 3

Shimomura¹

1997

Trans. Amer. Math. Soc.

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Abstract. In this paper, we try to compute the homotopy groups of the L 2 -localized Toda-Smith spectrum V (1) at the prime 3 by using the AdamsNovikov spectral sequence, and have almost done so. This computation involves non-trivial differentials d 5 and d 9 of the Adams-Novikov spectral sequence, different from the case p > 3. We also determine the homotopy groups of some L 2 -localized finite spectra relating to V (1). We further show some of the non-trivial differentials on elements relating so-called β-elements in the Adams-Novikov spectral sequence for π * (S 0 ). IntroductionLet L n be the Bousfield localization functor from the category of spectra to itself with respect to E(n) [19], and V (k) denote the Toda-Smith spectrum [26] with BP * -homology BP * /(p, v 1 , · · · , v n ), at each prime number p. Here BP and E(n) denote the Brown-Peterson and the Johnson-Wilson spectra with coefficient ringsn ] with |v n | = 2(p n − 1). The Toda-Smith spectrum V (k) is known to exist for k < 4 if and only if p > 2k [26], [20], and note that V (−1) = S 0 and V (0) is the mod p Moore spectrum. Determination of the homotopy groups of the L n -localized sphere spectrum is one of the key problems to understanding the category of L n -localized spectra. It is well known that π * (L 0 S 0 ) = Q. The homotopy groups π * (L 1 S 0 ) are determined by Ravenel [19] and π * (L 2 S 0 ) is determined for p > 3 by Yabe and the author [24]. Thus the next case will be π * (L 2 S 0 ) at the prime 3. At the prime p > 3, the homotopy groups π * (L 2 S 0 ) are obtained from π * (L 2 V (1)) by the v 1 -and p-Bockstein spectral sequences. Furthermore, at p > 3, the homotopy groups π * (L 2 S 0 ) are isomorphic to the E 2 -term of the Adams-Novikov spectral sequence, and they can be determined purely algebraically. Besides, π * (L 2 V (1)) is obtained from the cohomology of the Morava stabilizer algebra S 2 , which is computed by Ravenel (cf. [20]). Different from the case p > 3, the homotopy groups at the prime 3 are not isomorphic to the E 2 -term of the Adams-Novikov spectral sequence, and besides the E 2 -term of the Adams-Novikov spectral sequence for computing π * (L 2 V (1)) is expressed by the language of cohomology of groups [4] (cf. [3], [28]). Actually, Henn noticed a mistake in [20, Th. 6.3.23] and gave the correction of it in [4] from the viewpoint of cohomology of groups. In this paper, we provide another verification of this result using the cobar complex in Section 5.Received by the editors October 31, 1995. 1991 Mathematics Subject Classification. Primary 55Q45, 55Q10, 55Q52. The homotopy groups π * (L n V (n − 1)) are computed by Ravenel [20] for n ≤ 3 and p > n+1, in which case the Adams-Novikov spectral sequence collapses. So the next case in this sense is p = n + 1, whose case involves non-trivial Adams-Novikov differentials. When n = 1, this is a celebrated result of Mahowald (cf. [6], [7], [8]). We study the case n = 2 here in this point of view.In order to state our results, we introduce the notations:Our main r...

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“…At an odd prime it is possible [70] to analyze the classical Adams spectral sequence directly; there the map v 1 acts parallel with the vanishing line, so convergence of the localized spectral sequence is automatic. In the case of the mod p Moore spectrum there is one nontrivial differential, which can be computed by comparing the classical Adams spectral sequence with the Adams spectral sequence based on BP .…”

Section: The Image Of J In the Ehp -Sequencementioning

confidence: 99%