Equivariant Homotopy and Cohomology Theory

May, J. P.; Cole, Michael; Comezana, Gustavo; Costenoble, Steven R.; Elmendorf, Anthony; Greenlees, J. P. C.; Lewis, L; Piacenza, Robert; Triantafillou, Georgia; Waner, Stefan

doi:10.1090/cbms/091

Cited by 271 publications

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“…called the restriction and the transfer of M , respectively (see [May96,IX,[4][5]). Let X, Y be Z/2-spectra.…”

Section: Introductionmentioning

confidence: 99%

Equivariant Anderson Duality and Mackey Functor Duality

Ricka¹

2015

Glasgow Math. J.

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Abstract. We show that the Z/2-equivariant n th integral Morava Ktheory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrum KO as a corollary. The study of Z/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of RO(Z/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.Conventions: In this paper, F denotes the field with two elements. When considering the Steenrod algebra and the chromatic tower, the prime number is assumed to be p = 2. The category of abelian groups is denoted Ab. For E an object in the category of spectra (resp. Z/2-equivariant spectra), we denote E * (resp. E ⋆ ) the cohomology theory represented by E, and E * (resp. E ⋆ ) the homology theory represented by E. The homotopy of E is denoted E * (resp. E ⋆ ). Equivariant cohomology theories are graded over the orthogonal representation ring, thus ⋆ is an orthogonal representation of Z/2. We denote 1 the trivial one dimensional representation, and α the sign representation.

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“…called the restriction and the transfer of M , respectively (see [May96,IX,[4][5]). Let X, Y be Z/2-spectra.…”

Section: Introductionmentioning

confidence: 99%

Equivariant Anderson Duality and Mackey Functor Duality

Ricka¹

2015

Glasgow Math. J.

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“…The results above generalize directly to the equivariant setting of commutative S G -algebras and their modules [6,9,12], where G is a compact Lie group and S G is the sphere G-spectrum. Here, for a commutative S G -algebra R, we take R * = π * (R G ).…”

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confidence: 75%

“…The splittings of Theorem 4 give model theoretic refinements of splittings in equivariant stable homotopy theory that are discussed in [8, V] and [12,XVII §6]. Those sources describe splittings of homology and cohomology theories, and it is now apparent that these splittings arise from splittings of corresponding equivariant stable categories.…”

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confidence: 99%

Idempotents and Landweber exactness in brave new algebra

May

2001

Homology, Homotopy and Applications

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We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative S-algebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to M U -modules.In 1997, not long after [6] was written, I gave an April Fool's talk on how to prove that BP is an E ∞ ring spectrum or equivalently, in the language of [6], a commutative S-algebra. Unfortunately, the problem of whether or not BP is an E ∞ ring spectrum remains open. However, two interesting remarks emerged and will be presented here. One concerns splittings along idempotents and the other concerns the Landweber exact functor theorem.One of the nicest things in [6] is its one line proof that KO and KU are commutative S-algebras. This is an application of the following theorem [6, VIII.2.2], or rather the special case that follows. Theorem 1. Let R be a cell commutative S-algebra, A be a cell commutative Ralgebra, and M be a cell R-module. Then the Bousfield localization λ : A −→ A M ofA at M can be constructed as the inclusion of a subcomplex in a cell commutative R-algebra. In particular, the commutative R-algebra A M is a commutative S-algebra by neglect of structure.The cell assumptions can always be arranged by use of the cofibrant replacement constructions in [6], so they result in no loss of generality. The theorem specializes as follows to algebraic localizations at elements of R * = π * (R) [6, VIII.4.2]. The connective real K-theory spectrum ko is a commutative S-algebra by multiplicative infinite loop space theory [11], and KO is the localization ko[β −1 ] obtained

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“…We here give a bare bones summary of the construction of the stable homotopy category, referring to [16] and [11] for details and to [22] for a more leisurely exposition. We aim to give just enough of the basic definitional framework that the reader can feel comfortable with the ideas.…”

Section: Spectra and The Stable Homotopy Categorymentioning

confidence: 99%