The unstable Adams spectral sequence for generalized homology

Bendersky, Martin; Curtis, Edward B.; Miller, Haynes

doi:10.1016/0040-9383(78)90028-9

Cited by 64 publications

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“…Theories like M U or E(n) do not satisfy the hypothesis, but Landweber exact theories are treated in [BCM78,BT00].…”

Section: (F )} F ⊆Xmentioning

confidence: 99%

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Bauer

2011

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Abstract. Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E * to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework for studying the algebra of such functors, which I call formal plethories, in the case where E * is a Prüfer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.

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“…Theories like M U or E(n) do not satisfy the hypothesis, but Landweber exact theories are treated in [BCM78,BT00].…”

Section: (F )} F ⊆Xmentioning

confidence: 99%

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Bauer

2011

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“…Let M be the category of free, graded E * -modules. Drawing on the results of [5][6][7] and those of § § 2 and 3, we introduce a certain associated abelian category U. Our main theorem is the following.…”

Section: The Unstable Cobar Complex For E-theorymentioning

confidence: 99%

On the Coalgebraic Ring and Bousfield–kan Spectral Sequence for a Landweber Exact Spectrum

Bendersky

Hunton

2004

Proceedings of the Edinburgh Mathematical Society

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We construct a Bousfield-Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum E with unit and which is related to the homotopy groups of a certain unstable E completion X ∧ E of a space X. For E an S-algebra this completion agrees with that of the first author and Thompson. We also establish in detail the Hopf algebra structure of the unstable cooperations (the coalgebraic module) E * (E * ) for an arbitrary Landweber exact spectrum E, extending work of the second author with Hopkins and with Turner and giving basis-free descriptions of the modules of primitives and indecomposables. Taken together, these results enable us to give a simple description of the E 2 -page of the E-theory Bousfield-Kan spectral sequence when E is any Landweber exact ring spectrum with unit. This extends work of the first author and others and gives a tractable unstable Adams spectral sequence based on a vn-periodic theory for all n.

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“…In this section, we summarize the main results of [2], which gives the construction and main properties of the unstable Adams-Novikov spectral sequence. When a prime p is fixed, BP refers to the Brown-Peterson spectrum associated with p. For a space X, the (reduced) homology groups of X with coefficients in BP are denoted by H^(X; BP).…”

Section: Acknowledgementmentioning

confidence: 99%

“…The category of unstable F-comodules will be called ^ (91 is called jtf(U) in [2]). By construction ^ is an abelian category.…”

Section: )]) This Means That There Is a Map \L/: M-+u(m) Such That Tmentioning

confidence: 99%

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Some Calculations In the Unstable Adams-Novikov Spectral Sequence

Bendersky

1980

Publ. Res. Inst. Math. Sci.

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The unstable Adams-Novikov spectral sequence for a space X is a sequence of groups {E r (X)} 9 r = 2, 3,..., which converge to the homotopy groups of X, and whose £ 2 " term depends on the complex cobordism groups of X. We investigate this spectral sequence when X is the infinite special unitary group SU, or one of the finite groups SU(n), or when X is an odd sphere S 2 " +1 .The reader is referred to [2] for the construction and properties of the unstable Adams-Novikov spectral sequence. For some purposes, it is convenient to localize at a prime p, in which case the complex cobordism homology theory, based on the spectrum MU, is replaced by Brown-Peterson homology, based on the spectrum BP. We then have a useful spectral sequence with many of the properties of the stable Adams-Novikov spectral sequence. Namely, the nitrations are less than or equal to the filtrations in the unstable Adams spectral sequence based on mod-p homology. When X is a space for which H*(X; BP) is free over the coefficient ring n*(BP) and cofree as a coalgebra, then the E 2 -term is isomorphic to an Ext group in an abelian category (see §2; also [2, § 7]). Furthermore, this Ext group may be computed as the homology of an unstable cobar complex which we describe explicitly in Section 2. In particular, these considerations apply to the cases X = SU, X = SU(ri), or X = S 2n+1 .We first consider the situation where X is a p-local H-space with torsion-free homotopy and torsion-free homology. The results of Wilson [10] and the Communicated by N.

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The unstable Adams spectral sequence for generalized homology

Cited by 64 publications

References 13 publications

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On the Coalgebraic Ring and Bousfield–kan Spectral Sequence for a Landweber Exact Spectrum

Some Calculations In the Unstable Adams-Novikov Spectral Sequence

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